Before you start, make sure you know: Linear equations (Math 8 (Pre-Algebra))
A factory makes 600 phone chargers a day, and 5% come out defective. What is the probability that between 25 and 35 chargers are defective? This is a binomial situation, $B(600, 0.05)$. Adding up eleven terms like $\binom{600}{k}(0.05)^k(0.95)^{600-k}$ by hand would take forever. When $n$ is large, a normal (bell) curve gives an excellent answer with one table lookup. First we need to be fluent with the bell curve and the z-table. Then we use them on binomials.
The normal curve
Heights of students, the time a bus ride takes, the weight of bags of rice: when you measure many individuals, the values pile up in the middle and thin out symmetrically on both sides. That bell shape is the normal distribution.
Play with the mean and the standard deviation. Notice that $\mu$ slides the bell left and right, and $\sigma$ makes it wide and low or narrow and tall. The area stays $1$.
In the widget, $a$ is the mean and $b$ is the standard deviation.
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The weight of apples from a harvest follows a normal distribution with a mean of $150$ ounces and a standard deviation of $15$ ounces. If apples weighing between $135$…
The weight of apples from a harvest follows a normal distribution with a mean of $150$ ounces and a standard deviation of $15$ ounces. If apples weighing between $135$ and $165$ ounces are considered "type A apples", calculate the probability that a randomly chosen apple is type A.
Every context (heights, times, scores) has its own $\mu$ and $\sigma$, and nobody prints a table for every pair. So we use one reference curve.
Here is a piece of the table:
$z$
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
1.0
.8413
.8438
.8461
.8485
.8508
.8531
.8554
.8577
.8599
.8621
1.1
.8643
.8665
.8686
.8708
.8729
.8749
.8770
.8790
.8810
.8830
1.2
.8849
.8869
.8888
.8907
.8925
.8944
.8962
.8980
.8997
.9015
1.3
.9032
.9049
.9066
.9082
.9099
.9115
.9131
.9147
.9162
.9177
For example, $P(Z \le 1.24)$: row $1.2$, column $.04$, so $P(Z \le 1.24) = 0.8925$.
The table only gives areas to the left, but symmetry and the total area of $1$ give everything else.
Reading the table backward
Sometimes you know the area and need $z$. Find the area inside the table, then read the row and column labels. If $P(Z \le k) = 0.9066$, the table shows $0.9066$ in row $1.3$, column $.02$, so $k = 1.32$. If the area is less than $0.5$, $k$ is negative. For $P(Z \le k) = 0.1075$, look up $1 - 0.1075 = 0.8925$, which gives $1.24$, so $k = -1.24$.
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Given the standard normal distribution $Z \sim N(0, 1)$, use the probability table to calculate the cumulative area to the left of the following values. Graphically…
Given the standard normal distribution $Z \sim N(0, 1)$, use the probability table to calculate the cumulative area to the left of the following values. Graphically represent the situation in part a.
In a $Z \sim N(0, 1)$ distribution, we know that the total area under the curve is equal to $1$. Use the properties of symmetry and the complementary event to find the…
In a $Z \sim N(0, 1)$ distribution, we know that the total area under the curve is equal to $1$. Use the properties of symmetry and the complementary event to find the following probabilities:
Find the value of the abscissa $k$ (the value of the variable on the horizontal axis) in a standard normal distribution $Z \sim N(0, 1)$ knowing that: a)…
Find the value of the abscissa $k$ (the value of the variable on the horizontal axis) in a standard normal distribution $Z \sim N(0, 1)$ knowing that:
How do we use the $N(0,1)$ table for a variable like $X \sim N(35, 6)$? We re-measure $X$ in "standard deviations from the mean."
Why does it work? Subtracting $\mu$ moves the center of the bell to $0$, and dividing by $\sigma$ rescales the spread to $1$. The area under the curve between two values does not change.
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A study on the time 16-year-old teenagers spend using a well-known social network per day indicates that it follows a normal distribution with a mean of $\mu = 120$…
A study on the time 16-year-old teenagers spend using a well-known social network per day indicates that it follows a normal distribution with a mean of $\mu = 120$ minutes and a standard deviation of $\sigma = 20$ minutes.
If we choose a young person at random, calculate the probability that they use the social network for less than $150$ minutes. To do this, standardize the variable $X$ and use the $N(0, 1)$ table.
The scores of an Applied Mathematics for Social Sciences exam follow a $N(5,2; 1,5)$ distribution. If a score higher than $8$ is required to obtain a special mention,…
The scores of an Applied Mathematics for Social Sciences exam follow a $N(5,2; 1,5)$ distribution. If a score higher than $8$ is required to obtain a special mention, what is the probability of a student achieving it?
Hint: Remember that $P(X > k) = 1 - P(X \le k)$ after standardizing.
The time that high school juniors dedicate daily to doing homework follows a normal distribution with a mean of $\mu = 115$ minutes and a standard deviation of…
The time that high school juniors dedicate daily to doing homework follows a normal distribution with a mean of $\mu = 115$ minutes and a standard deviation of $\sigma = 25$ minutes. Calculate the probability that a randomly chosen student:
A $75$ on a hard exam might be better than an $85$ on an easy one. To compare values from different distributions, compare their z-scores: the larger z-score is better relative to its own group.
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Two friends, Marcus and Sarah, study at different schools. At Marcus's school, the mean score on the Economics exam was $6,2$ with a standard deviation of $1,1$. At…
Two friends, Marcus and Sarah, study at different schools. At Marcus's school, the mean score on the Economics exam was $6,2$ with a standard deviation of $1,1$. At Sarah's, the mean was $6,8$ with a standard deviation of $1,5$. If Marcus obtained a $7,5$ and Sarah a $8,2$, and assuming that both populations follow a normal distribution, determine which of the two obtained a better relative score compared to their group.
"How tall do you have to be to be in the tallest $10\%$?" Now the probability is given and you need the value $x$. This is the table in reverse, followed by undoing the standardizing.
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A gaming peripherals brand has determined that the response time of its flagship mouse follows a $N(18, 4)$ milliseconds distribution. The marketing department wants to…
A gaming peripherals brand has determined that the response time of its flagship mouse follows a $N(18, 4)$ milliseconds distribution. The marketing department wants to promote that their product is so fast that only $15\%$ of competitors' mice (which follow the same distribution) exceed a critical response time $k$. Find the value of this time $k$.
The battery life of a smartphone model follows a $N(42, 6)$ hours distribution. The company wants to offer a replacement warranty if the battery lasts less than a…
The battery life of a smartphone model follows a $N(42, 6)$ hours distribution. The company wants to offer a replacement warranty if the battery lasts less than a certain time. What maximum duration should the company guarantee if it is only willing to replace $2,5\%$ of the batteries sold?
In a prestigious esports tournament, the score obtained by the participants follows a normal distribution with a mean of 2500 points and a standard deviation of 400…
In a prestigious esports tournament, the score obtained by the participants follows a normal distribution with a mean of 2500 points and a standard deviation of 400 points. The organization decides to award the "Elite Mention" only to the 2.5% of players with the highest scores.
a) From what score is this mention received?
b) If a group of friends has scores that lie in the central interval comprising 95% of the players in the tournament, between which two values do their scores range?
Sometimes the unknown is not a probability or a cutoff but a parameter of the distribution. You still find $z_0$ from the area. But now $z_0 = \dfrac{x - \mu}{\sigma}$ becomes an equation in $\mu$ and/or $\sigma$.
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It is known that the weight of coffee packets from a certain factory follows a normal distribution. After a quality control check, it is observed that $5\%$ of the…
It is known that the weight of coffee packets from a certain factory follows a normal distribution. After a quality control check, it is observed that $5\%$ of the packets weigh less than $242$ ounces and $10\%$ weigh more than $255$ ounces. Find the mean $\mu$ and the standard deviation $\sigma$ of this population.
It is known that the charging time of the batteries of a new model of electric car follows a normal distribution $N(\mu, \sigma)$. After carrying out a market study, it…
It is known that the charging time of the batteries of a new model of electric car follows a normal distribution $N(\mu, \sigma)$. After carrying out a market study, it is observed that 15% of the vehicles take less than 45 minutes to charge completely, while only 20% of the vehicles require more than 75 minutes.
Calculate the value of the mean $\mu$ and the standard deviation $\sigma$ of this distribution. Then, determine the probability that a randomly chosen vehicle takes between 50 and 60 minutes to charge.
A courier company uses drones for its deliveries in urban centers. It has been proven that the flight time of these drones until the battery runs out follows a normal…
A courier company uses drones for its deliveries in urban centers. It has been proven that the flight time of these drones until the battery runs out follows a normal distribution with a mean of 42 minutes. If 97.72% of the drones fly for more than 30 minutes, what is the probability that a randomly chosen drone runs out of battery between 35 and 45 minutes of flight?
A binomial variable $X \sim B(n, p)$ counts successes in $n$ independent trials, each with success probability $p$. Let $q = 1 - p$. Its mean is $np$ and its standard deviation is $\sqrt{npq}$. Make $n$ large and the histogram of $X$ looks more and more like a bell.
Why the condition? If $p$ is close to $0$ or $1$ and $n$ is not huge, the binomial histogram piles up against $0$ or $n$ and is lopsided. A symmetric bell cannot match it.
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A basketball free-throw shooter has a probability of success of $p = 0,8$. If they take $n = 100$ shots, we want to know if we can approximate the binomial variable $X$…
A basketball free-throw shooter has a probability of success of $p = 0,8$. If they take $n = 100$ shots, we want to know if we can approximate the binomial variable $X$ (number of successful shots) by a normal distribution.
a) Check if the conditions $n \cdot p \ge 5$ and $n \cdot q \ge 5$ are met.
b) If so, state what the parameters $\mu$ and $\sigma$ of the new normal distribution $X'$ would be.
A binomial takes only whole-number values $0, 1, 2, \dots$. A normal is continuous. For a single value, the normal gives $P(X' = 65) = 0$, which is useless. The fix is to let each whole number $k$ occupy the interval from $k - 0.5$ to $k + 0.5$. That is exactly the width of its bar in the histogram. Some problems call this Yates's continuity correction.
If a problem tells you to skip the correction (to see how it behaves), just standardize the number exactly as it is written.
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A fair coin ($p = 0,5$) is flipped $400$ times. Calculate the probability of getting fewer than $190$ heads using the normal approximation to the binomial (without…
A fair coin ($p = 0,5$) is flipped $400$ times. Calculate the probability of getting fewer than $190$ heads using the normal approximation to the binomial (without applying the continuity correction yet).
Step 1: Calculate the mean $\mu$ and the standard deviation $\sigma$.
Step 2: Standardize the value $X = 190$ and look it up in the table.
On a streaming platform, $15\%$ of users cancel their subscription before the end of the trial month. If $400$ new users have signed up this month, calculate, using the…
On a streaming platform, $15\%$ of users cancel their subscription before the end of the trial month. If $400$ new users have signed up this month, calculate, using the normal approximation with continuity correction, the probability that exactly $65$ users unsubscribe.
A biased coin is flipped 500 times, where the probability of getting heads is 0.4. Calculate the probability of getting: a) Exactly 200 heads. b) More than 215 heads. c)…
A biased coin is flipped 500 times, where the probability of getting heads is 0.4. Calculate the probability of getting:
a) Exactly 200 heads.
b) More than 215 heads.
c) Fewer than 185 heads.
Note: Use the normal approximation with Yates's continuity correction.
A study on recycling habits states that $35\%$ of households in a city separate plastic correctly. If a random sample of $120$ households is selected: a) Justify why…
A study on recycling habits states that $35\%$ of households in a city separate plastic correctly. If a random sample of $120$ households is selected:
a) Justify why this experiment can be approximated by a normal distribution.
b) Calculate the probability that, in this sample, between $40$ and $50$ households (inclusive) separate plastic.
A multiple-choice exam consists of $80$ questions, each with $4$ answer options of which only one is correct. If a student answers all the questions completely at…
A multiple-choice exam consists of $80$ questions, each with $4$ answer options of which only one is correct. If a student answers all the questions completely at random:
a) Determine the parameters of the normal distribution that approximates this variable.
b) Calculate the probability that they get at least $25$ correct answers.
A study on reading habits in teenagers reveals that 70% of young people aged between 15 and 16 prefer digital format over paper. If we select a random sample of 400…
A study on reading habits in teenagers reveals that 70% of young people aged between 15 and 16 prefer digital format over paper. If we select a random sample of 400 teenagers:
a) Justify whether it is possible to approximate the distribution of the variable "number of teenagers who prefer digital format" using a normal distribution.
b) Calculate, using the previous approximation and applying the continuity correction, the probability that between 270 and 290 teenagers (inclusive) prefer the digital format.
Some problems ask what outcomes are typical, or what is the largest count you can expect with a given probability. Both use the same normal tools.
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5% of the parts manufactured by a high-precision machine are defective. If they are packed in boxes of 200 units, calculate the probability that in a box: a) There are…
5% of the parts manufactured by a high-precision machine are defective. If they are packed in boxes of 200 units, calculate the probability that in a box:
a) There are exactly 10 defective parts.
b) There are more than 15 defective parts.
c) The number of defective parts is less than or equal to 5.
Argue whether the results would be significantly different if the continuity correction were not applied.
In a multiple-choice exam with 100 questions, each with 4 answer options (only one correct), a student answers all the questions at random. a) What is the probability…
In a multiple-choice exam with 100 questions, each with 4 answer options (only one correct), a student answers all the questions at random.
a) What is the probability that they get more than 30 questions right?
b) If at least 40 correct answers are needed to pass the exam, what is the probability that the student passes purely by chance?
c) Determine the interval $(np - 2\sigma, np + 2\sigma)$ and interpret its meaning in this context.
A gym chain knows that 20% of its registered members never visit the facilities. If there are 600 registered members in a specific gym: a) Calculate the probability that…
A gym chain knows that 20% of its registered members never visit the facilities. If there are 600 registered members in a specific gym:
a) Calculate the probability that the number of members who never visit is between 110 and 135 inclusive.
b) If the gym decides to launch a motivation campaign for these members if the number of "inactive" members exceeds 140, what is the probability that they will have to launch this campaign?
c) What is the maximum number of inactive members expected to be found with a probability of 90%?
Back to the start: For the charger factory, $X \sim B(600, 0.05)$ with $np = 30 \ge 5$ and $nq = 570 \ge 5$. Then $\mu = 30$ and $\sigma = \sqrt{600 \cdot 0.05 \cdot 0.95} = \sqrt{28.5} \approx 5.34$. With the continuity correction, $P(25 \le X \le 35) \to P(24.5 \le X' \le 35.5)$. The z-scores are $\pm\dfrac{5.5}{5.34} \approx \pm 1.03$. So the probability is $2(0.8485) - 1 = 0.6970$, about $69.7\%$.