Bernoulli trials
Orla takes three frees in a camogie match. She scores each one with probability $0.6$, and one free does not affect the next. What is the probability that she scores all three? What is the probability that she scores exactly one? By the end of this page you will be able to answer both, and we will come back to Orla in the summary.
What is a Bernoulli trial?
Lots of situations in real life have just two possible results: a free is scored or missed, a bus is on time or late, a seed germinates or does not. If the chance stays the same every time, and one attempt does not affect the next, we call each attempt a Bernoulli trial.
"Success" is just a label for the outcome we are counting. If we are counting faulty items, then a faulty item is a "success".
Toss the coin a few times. Each toss is a Bernoulli trial with $p = \frac{1}{2}$ and $q = \frac{1}{2}$.
The probability that a bus in Limerick arrives on time is $0.8$, independently each day. Calculate the probability that it is on time on each of three consecutive days.
The probability that a bus in Limerick arrives on time is $0.8$, independently each day. Calculate the probability that it is on time on each of three consecutive days.
Several trials in a row
Because the trials are independent, we multiply the probabilities along a path. This is the rule for two independent events both occurring, extended to three trials.
A fair coin is tossed 3 times. Calculate the probability of getting exactly one head.
A fair coin is tossed 3 times. Calculate the probability of getting exactly one head.
Exactly some successes
What if the successes can happen in any order? Then we list all the sequences that give the result we want, find the probability of each one, and add them (the sequences are mutually exclusive).
Here is the full list for three trials, with $S$ for success and $F$ for failure:
| Number of successes | Sequences | Number of sequences |
|---|---|---|
| 0 | FFF | 1 |
| 1 | SFF, FSF, FFS | 3 |
| 2 | SSF, SFS, FSS | 3 |
| 3 | SSS | 1 |
The total is $1 + 3 + 3 + 1 = 8 = 2^3$ sequences. Every sequence with the same number of successes has the same probability, so we only need to calculate one and multiply by the number of sequences.
A penalty taker in a Gaelic football club scores with probability $\frac{3}{4}$ each attempt, independently. Find the probability that: a) the first miss occurs on the…
A penalty taker in a Gaelic football club scores with probability $\frac{3}{4}$ each attempt, independently. Find the probability that:
- a) the first miss occurs on the second attempt;
- b) he scores exactly two out of three penalties;
- c) he scores at most one out of three penalties.
A basketball player from Killarney scores a free throw with probability $0.7$, independently each time. She takes 3 free throws. a) Calculate the probability that she…
A basketball player from Killarney scores a free throw with probability $0.7$, independently each time. She takes 3 free throws.
- a) Calculate the probability that she scores exactly two.
- b) Calculate the probability that she scores at least one.
"At least one": use the complement
"At least one success in 3 trials" means 1, 2 or 3 successes. Adding three cases is slow. There is only one case that is not included: no successes at all. So we use the complement:
The same idea works for "at least two" in 3 trials: it means 2 or 3 successes, so you add those two cases. You can also do $1 - P(0) - P(1)$. Choose whichever involves fewer cases.
A machine in a factory in Waterford produces a faulty item with probability $0.1$, independently of other items. a) Find the probability that the first faulty item is…
A machine in a factory in Waterford produces a faulty item with probability $0.1$, independently of other items.
- a) Find the probability that the first faulty item is the $n$th item produced, in terms of $n$.
- b) Find the probability that the first faulty item is the fifth item or later.
- c) Three items are selected. Find the probability that at least two are faulty.
The first success on the nth trial
Sometimes we repeat the trial until the first success. For the first success to happen on trial $n$, the first $n - 1$ trials must all be failures, and then trial $n$ must be a success.
There is only one sequence here (the order is fixed), so we do not multiply by anything extra.
A fair die is rolled until a 6 appears. Calculate the probability that the first 6 occurs on the third roll.
A fair die is rolled until a 6 appears. Calculate the probability that the first 6 occurs on the third roll.
The probability that a seed sown in a Wexford greenhouse germinates is $0.9$. Seeds germinate independently. a) Find the probability that the first seed to fail to…
The probability that a seed sown in a Wexford greenhouse germinates is $0.9$. Seeds germinate independently.
- a) Find the probability that the first seed to fail to germinate is the fourth seed sown.
- b) Find the probability that, of three seeds, exactly one fails to germinate.
Seán rolls a fair die repeatedly until he gets a 6. a) Show that the probability the first 6 occurs on roll $n$ is $\left(\frac{5}{6}\right)^{n-1}\cdot\frac{1}{6}$. b)…
Seán rolls a fair die repeatedly until he gets a 6.
- a) Show that the probability the first 6 occurs on roll $n$ is $\left(\frac{5}{6}\right)^{n-1}\cdot\frac{1}{6}$.
- b) Find the probability that the first 6 occurs within the first three rolls.
- c) Find the least value of $n$ such that the probability of getting at least one 6 in $n$ rolls exceeds $0.9$.
Probability distribution and expected number of successes
We can put all the possibilities for the number of successes into a probability distribution: a table of each value $x$ and its probability $P(x)$. The probabilities add up to 1.
A multiple-choice quiz at a school in Sligo has 3 questions, each with 4 options of which exactly one is correct. A student guesses every answer. a) Find the probability…
A multiple-choice quiz at a school in Sligo has 3 questions, each with 4 options of which exactly one is correct. A student guesses every answer.
- a) Find the probability distribution for the number of correct answers.
- b) Calculate the probability that the student gets a pass, defined as at least 2 correct answers.
- c) Calculate the expected number of correct answers and compare it with $3 \times \frac{1}{4}$.
Summary
-
A Bernoulli trial has two outcomes (success $p$, failure $q = 1 - p$), a fixed probability and independent repeats.
-
Multiply probabilities along a sequence: for example SSF has probability $p^2 q$.
-
For exactly $k$ successes in 3 trials, list the sequences, find the probability of one and multiply by the number of sequences (1, 3, 3, 1 for 0, 1, 2, 3 successes).
-
At least one success: $1 - q^n$.
-
First success on trial $n$: $q^{n-1}p$. First success on trial $n$ or later: $q^{n-1}$.
-
Expected number of successes in $n$ trials: $np$.
Back to the start: Orla scores all three frees with probability $0.6^3 = 0.216$. She scores exactly one with probability $3 \times 0.6 \times 0.4^2 = 0.288$.