Probability
Coach tells Aaliyah that she makes about 70% of her free throws. In tonight's warm-up she makes only 5 of her first 10. Is the coach wrong? To answer, you need a number that says how likely something is, and a way to tell when you can trust it. That number is called probability.
Random experiments and sample spaces
Some experiments always turn out the same way. Others do not.
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Deterministic: if you repeat it under the same conditions, you always get the same result. Example: adding $2 + 2$ on a calculator.
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Random: you know the possible results, but you cannot predict which one will happen next. Example: drawing a name from a hat.
Classify the following experiments as random or deterministic, briefly justifying your answer: a) Observing whether it will rain tomorrow in your city. b) Measuring the…
Classify the following experiments as random or deterministic, briefly justifying your answer:
- a) Observing whether it will rain tomorrow in your city.
- b) Measuring the time it takes for a stone to fall from a second floor.
- c) Noting down the winning number in the next lottery draw.
- d) Knowing the temperature at which distilled water boils at sea level.
You have the following wheel of fortune divided into eight equal sectors. If you spin it and note the resulting color: a) Write down the sample space $E$. b) State two…
You have the following wheel of fortune divided into eight equal sectors. If you spin it and note the resulting color:
- a) Write down the sample space $E$.
- b) State two elementary events.
- c) Write down the elements of the compound event $A$: "landing on a primary color" (considering red, blue, and yellow as such).
Experimental probability and the Law of Large Numbers
When outcomes are not symmetric (a thumbtack, a bottle flip, a weighted die), you cannot just count sides. You experiment. Repeat the experiment $N$ times. If the event $A$ happens $f$ times, its relative frequency is
$$\frac{f}{N}$$This is also called the experimental probability of $A$.
Here is Aaliyah's data from the hook, collected over a whole season:
| Shots | 10 | 50 | 200 | 1,000 |
|---|---|---|---|---|
| Made | 5 | 36 | 138 | 702 |
| Relative frequency | 0.5 | 0.72 | 0.69 | 0.702 |
With few shots the relative frequency jumps around. With many shots it settles near one value. Try it with a die:
Experimental results will not match a claimed probability perfectly. If a spinner is supposed to land on red 25% of the time and gives 238 reds in 1,000 spins, the relative frequency is $0.238$. That is close to $0.25$, so the claim is believable. A gap of a few percentage points is normal for that many trials.
A pushpin has been dropped many times to observe whether it lands point up or point down. The results are recorded in the following table: | Drops | $50$ | $100$ | $500$…
A pushpin has been dropped many times to observe whether it lands point up or point down. The results are recorded in the following table:
| Drops | $50$ | $100$ | $500$ | $1000$ |
|---|---|---|---|---|
| Point up | $18$ | $32$ | $155$ | $312$ |
- a) Calculate the relative frequency of the event "landing point up" for each column.
- b) Based on the Law of Large Numbers, what value would you assign to the probability of the pushpin landing point up?
A video game developer is testing a new reward system. They have programmed a virtual chest that, when opened, uses an algorithm to decide the prize. According to the…
A video game developer is testing a new reward system. They have programmed a virtual chest that, when opened, uses an algorithm to decide the prize. According to the data from the last $2000$ chests opened by players, the following results have been obtained:
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Common item: $1240$ times.
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Rare item: $680$ times.
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Legendary item: $80$ times.
- a) Calculate the empirical probability (relative frequency) of getting a legendary item.
- b) If the developer states that the theoretical probability of getting a rare or legendary item is $40\%$, do you think the algorithm is working correctly according to the Law of Large Numbers? Justify your answer by comparing the probabilities.
- c) If a player opens two chests in a row, what is the probability that both contain a common item? (Assume the events are independent).
Theoretical probability
Sometimes you know the probability without experimenting, because all outcomes are equally likely: a fair die, a well-shuffled deck, identical marbles in a bag.
Try a bag of 6 red and 6 blue marbles. How many draws do you need before the relative frequencies get close to the theoretical $\frac{1}{2}$?
When you count favorable outcomes, write them out. For "a prime number" on a numbered die, remember that a prime has exactly two factors, so $1$ is not prime and $2$ is.
In a sports bag, there are $6$ soccer balls, $4$ basketballs, and $2$ handballs. If we take out a ball at random to start training, calculate the probability of the…
In a sports bag, there are $6$ soccer balls, $4$ basketballs, and $2$ handballs. If we take out a ball at random to start training, calculate the probability of the following events applying Laplace's Rule:
- a) $P(\text{football})$
- b) $P(\text{basketball})$
- c) $P(\text{not a handball})$
A board game uses a dodecahedron-shaped die (12 equal faces) numbered from $1$ to $12$. Calculate the probability of getting: a) A prime number. b) A multiple of $4$. c)…
A board game uses a dodecahedron-shaped die (12 equal faces) numbered from $1$ to $12$. Calculate the probability of getting:
- a) A prime number.
- b) A multiple of $4$.
- c) A number greater than $9$.
We draw a card from a Spanish deck of $40$ cards. Find the probability of the following events: a) Drawing the King of Cups. b) Drawing an "ace". c) Drawing a card from…
We draw a card from a Spanish deck of $40$ cards. Find the probability of the following events:
- a) Drawing the King of Cups.
- b) Drawing an "ace".
- c) Drawing a card from the suit of Coins.
- d) Drawing a "face card" (Jack, Knight or King).
Probability scale and complements
Probabilities live between $0$ and $1$. You can write them as fractions, decimals, or percents.
A weather service announces that the probability of it snowing in the mountains tomorrow is $0,35$. a) What is the probability that it does not snow? b) If event $A$ is…
A weather service announces that the probability of it snowing in the mountains tomorrow is $0,35$.
- a) What is the probability that it does not snow?
- b) If event $A$ is "it snows", what is the event of it not snowing called with respect to $A$?
- c) Express both probabilities as a percentage.
In a playlist on a music streaming service, Luke has 40 songs of different genres: 12 are Pop, 8 are Rock, 15 are Trap, and 5 are Indie. If Luke selects the "shuffle"…
In a playlist on a music streaming service, Luke has 40 songs of different genres: 12 are Pop, 8 are Rock, 15 are Trap, and 5 are Indie. If Luke selects the "shuffle" option, calculate the probability that the first song to play:
- a) Is Trap.
- b) Is not Indie.
- c) Is Pop or Rock.
"Or" events that overlap
In math, "$A$ or $B$" means $A$, or $B$, or both. You may see it written $A \cup B$ (the union). "$A$ and $B$" means both happen at once, written $A \cap B$ (the intersection).
When $A$ and $B$ can happen together, adding $P(A) + P(B)$ counts the overlap twice. So we subtract it once.
A Venn diagram shows the overlap clearly. Suppose 40 students are in a club: 22 play basketball, 15 run track, and 11 do neither. How many do both?
We roll a role-playing game die in the shape of a dodecahedron (12 faces numbered from 1 to 12). We define the following events: $A = \text{"Getting a multiple of 3"}$…
We roll a role-playing game die in the shape of a dodecahedron (12 faces numbered from 1 to 12). We define the following events:
$A = \text{"Getting a multiple of 3"}$
$B = \text{"Getting a number greater than 7"}$
Calculate the following probabilities:
- a) $P(A)$
- b) $P(B)$
- c) $P(A \cap B)$
- d) $P(A \cup B)$
A lottery drum contains 100 balls numbered from 00 to 99. a) What is the probability that the drawn ball ends in 7? b) What is the probability that it is a number…
A lottery drum contains 100 balls numbered from 00 to 99.
- a) What is the probability that the drawn ball ends in 7?
- b) What is the probability that it is a number between 20 and 35 (inclusive)?
- c) What is the probability that it is neither an even number nor a multiple of 10?
A study has been carried out on the use of social media in a middle school of $300$ students. The results are as follows: $180$ students use Instagram. $130$ students…
A study has been carried out on the use of social media in a middle school of $300$ students. The results are as follows:
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$180$ students use Instagram.
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$130$ students use TikTok.
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$40$ students do not use either of the two networks.
If we choose a student at random, calculate the probability that they:
- a) Use both social networks.
- b) Use Instagram but not TikTok.
- c) Given that the student uses TikTok, what is the probability that they also use Instagram?
Two-way tables and "given that"
A two-way table sorts the same group by two categories. Here is a survey of 50 students about their favorite lunch:
You can find ordinary probabilities from the totals: $P(\text{tacos}) = \dfrac{27}{50}$. And "and" probabilities come from a single cell: $P(\text{8th grade and tacos}) = \dfrac{16}{50}$.
Sometimes you get extra information, such as "given that the student is in 8th grade." Now the group shrinks to that row, and the new total is $25$.
In a 10th grade class, a survey was carried out on the preferred social network for following influencers. The results are shown in the following table: | | TikTok |…
In a 10th grade class, a survey was carried out on the preferred social network for following influencers. The results are shown in the following table:
| TikTok | TOTAL | ||
|---|---|---|---|
| Boys | 12 | 6 | 18 |
| Girls | 10 | 12 | 22 |
| TOTAL | 22 | 18 | 40 |
If we choose a student at random, calculate the probability that they:
- a) Are a girl.
- b) Prefer Instagram.
- c) Are a boy who prefers TikTok.
- d) Given that they are a boy, prefer Instagram.
Compound experiments: lists, tables, and trees
A compound experiment has more than one stage, like flipping a coin and rolling a die. To find its sample space, list outcomes systematically. A tree diagram does this automatically: each path from left to right is one outcome.
For two dice, a table works better than a tree. Rows are the first die and columns are the second. Each cell is one of the $36$ equally likely outcomes. Here the cells show the sum:
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
A sum of 7 appears $6$ times, so $P(\text{sum is 7}) = \dfrac{6}{36} = \dfrac{1}{6}$. A sum of 2 appears only once, so $P(\text{sum is 2}) = \dfrac{1}{36}$. You can test this with the two-dice experiment:
We toss two $\$1…
We toss two $\$1$ coins in the air and observe the result of the upper faces.
- a) Draw the tree diagram for this experiment.
- b) Write down the complete sample space.
- c) What is the probability of getting exactly two "tails"?
A coin is flipped and a spinner divided into 5 equal sectors numbered from 1 to 5 is spun. a) Write down the sample space of this compound experiment. b) What is the…
A coin is flipped and a spinner divided into 5 equal sectors numbered from 1 to 5 is spun.
- a) Write down the sample space of this compound experiment.
- b) What is the probability of getting "Heads" and an even number?
- c) What is the probability of getting "Tails" and a number less than 5?
In a board game, two six-sided dice are rolled. A player wins if the positive difference between the scores on the two dice is a prime number. a) Construct a two-way…
In a board game, two six-sided dice are rolled. A player wins if the positive difference between the scores on the two dice is a prime number.
- a) Construct a two-way table to show all the possible differences.
- b) What is the probability of winning the game?
- c) Is it more likely to get a difference that is a prime number or a difference that is a multiple of $3$? Justify your answer by calculating both probabilities.
Independent events and tree diagrams
Two events are independent if one happening does not change the probability of the other: two coin flips, two separate buses, shots with a fresh start each time. When outcomes are not equally likely, put the probabilities on the branches.
Aaliyah makes 70% of her free throws and each shot is independent. Here are two shots:
When the same result can occur on several paths, count them all.
With more questions, list the paths systematically. For example, there are $6$ ways to place exactly two right answers among four questions. Stay organized so you do not miss one or repeat one.
A student has found that the bus to school arrives late 4 out of 10 days due to traffic. If they have to catch two different buses that operate independently to get to…
A student has found that the bus to school arrives late 4 out of 10 days due to traffic. If they have to catch two different buses that operate independently to get to school, calculate:
- a) The probability that both buses arrive on time.
- b) The probability that at least one of the two buses arrives late.
Two esports teams, the "Pixels" and the "Lag-Killers", are playing the final of a best-of-three tournament (the first to get two wins is the winner). From previous…
Two esports teams, the "Pixels" and the "Lag-Killers", are playing the final of a best-of-three tournament (the first to get two wins is the winner). From previous statistics, we know that in each individual match, the "Pixels" have a $P(P) = 0,6$ probability of winning, while the "Lag-Killers" have $P(L) = 0,4$. There is no possibility of a tie.
Use a tree diagram to calculate:
- a) The probability that the "Pixels" win the tournament in just two matches.
- b) The probability that the tournament is decided in the third match.
- c) The total probability that the "Lag-Killers" make a comeback and win the tournament.
A multiple-choice exam consists of $4$ questions with $3$ answer options each, of which only one is correct. A student who has not studied decides to answer all the…
A multiple-choice exam consists of $4$ questions with $3$ answer options each, of which only one is correct. A student who has not studied decides to answer all the questions at random.
- a) What is the probability that they do not get any questions right?
- b) What is the probability that they get exactly two questions right?
- c) If they need to get at least $3$ questions right to pass, what is the probability of them passing the exam?
Drawing without replacement
Pulling items out of a bag and not putting them back makes the events dependent: the first draw changes what is left, so it changes the second probability.
A gumball machine has 3 green and 2 yellow gumballs. You take two without putting any back:
Notice the second-stage branches. After a green is removed, there are only $4$ gumballs left and $2$ of them are green, so $\frac{2}{4}$. After a yellow is removed, there are $4$ left and $3$ are green, so $\frac{3}{4}$.
When the problem says "given that the first was green, what is the chance the second is yellow?", you are being asked for a branch of the second stage: $\frac{2}{4}$. You can also read it from the tree.
For "at least one" with three or more draws, use the complement again. For example, "at least one black pick" is $1 - P(\text{no black picks})$, and you multiply the shrinking fractions for "no black".
In a sports bag, there are 5 white shirts and 3 black shirts. If we draw two shirts at random and consecutively without returning the first one to the bag, calculate the…
In a sports bag, there are 5 white shirts and 3 black shirts. If we draw two shirts at random and consecutively without returning the first one to the bag, calculate the probability that:
- a) Both are white.
- b) They are different colors.
- c) The second is black, given that the first was white.
In a music box, there are $12$ guitar picks: $5$ are red, $4$ are blue, and $3$ are black. If we draw three picks from the box, one after the other and without…
In a music box, there are $12$ guitar picks: $5$ are red, $4$ are blue, and $3$ are black. If we draw three picks from the box, one after the other and without replacement, calculate:
- a) The probability that all three picks are the same color.
- b) The probability that the first is red, the second is blue, and the third is black, in that exact order.
- c) The probability that at least one of the drawn picks is black.
When the first stage changes the second
Sometimes the second stage's probabilities depend on what happened first, even though nothing is "removed." The tree handles it: each second-stage branch gets the probability that fits its own first-stage result. To find the chance of an event that can happen along several paths, add those paths.
The same idea works when an object moves between containers. If you move a seed from pot $A$ into pot $B$ before drawing from $B$, what $B$ contains depends on what moved. So split into the two cases, give each case its own second-stage probabilities, then add the paths that give the event you want.
In a video game tournament, a player has a $0,6$ probability of winning the first match. If they win the first one, their confidence increases and the probability of…
In a video game tournament, a player has a $0,6$ probability of winning the first match. If they win the first one, their confidence increases and the probability of winning the second is $0,7$. However, if they lose the first one, the probability of winning the second drops to $0,3$.
Use a tree diagram to help you calculate the probability that they:
- a) Win both matches.
- b) Win at least one of the two matches.
In a school reforestation project, we have two pots with seeds. Pot $A$ contains $4$ oak seeds and $6$ pine seeds. Pot $B$ contains $5$ oak seeds and $3$ pine seeds. We…
In a school reforestation project, we have two pots with seeds. Pot $A$ contains $4$ oak seeds and $6$ pine seeds. Pot $B$ contains $5$ oak seeds and $3$ pine seeds.
We carry out the following experiment: we draw a seed at random from pot $A$ and move it to pot $B$ without looking at it. Then, we draw a seed from pot $B$.
- a) What is the probability that the seed drawn from $B$ is an oak seed?
- b) What is the probability that both drawn seeds (the one we moved from $A$ to $B$ and the one from $B$) are of the same species?
Geometric probability
Sometimes the outcomes are points in a region, not items you can count. If every point is equally likely, the probability depends on area:
$$P(\text{landing in the target}) = \frac{\text{area of the target}}{\text{area of the whole region}}$$Why? A target with twice the area catches twice as many of the equally likely points, so it is twice as likely to be hit.
A courier company uses drones to deliver parcels in a rural area. The landing area is a square with a side length of $10$ feet. Due to the inaccuracy of the wind sensor,…
A courier company uses drones to deliver parcels in a rural area. The landing area is a square with a side length of $10$ feet. Due to the inaccuracy of the wind sensor, the parcel can land at any point in the square with equal probability. Inside this square, a safety circle with a radius of $3$ feet has been painted. If the parcel lands outside the circle, the drone must land again for safety.
- a) Calculate the probability that a parcel lands inside the safety circle on the first attempt.
- b) If the drone makes two independent deliveries, what is the probability that at least one of them lands outside the circle? (Use $\pi \approx 3,14$).
Back to the start: Aaliyah's 5 out of 10 is a relative frequency of $0.5$. With only 10 shots, that can easily land far from her true $0.7$. By 1,000 shots her relative frequency was $0.702$, so the coach is not wrong. Ten shots are simply too few to judge. And now you can go further: the probability that she misses both of two shots is $0.3 \cdot 0.3 = 0.09$.