Exercises: Bernoulli trials

A fair coin is tossed 3 times. Calculate the probability of getting exactly one head.

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.

A fair die is rolled until a 6 appears. Calculate the probability that the first 6 occurs on the third roll.

A basketball player from Killarney scores a free throw with probability $0.7$, independently each time. She takes 3 free throws.

1. a) Calculate the probability that she scores exactly two.
2. b) Calculate the probability that she scores at least one.

The probability that a seed sown in a Wexford greenhouse germinates is $0.9$. Seeds germinate independently.

1. a) Find the probability that the first seed to fail to germinate is the fourth seed sown.
2. b) Find the probability that, of three seeds, exactly one fails to germinate.

A penalty taker in a Gaelic football club scores with probability $\frac{3}{4}$ each attempt, independently. Find the probability that:

1. a) the first miss occurs on the second attempt;
2. b) he scores exactly two out of three penalties;
3. c) he scores at most one out of three penalties.

A machine in a factory in Waterford produces a faulty item with probability $0.1$, independently of other items.

1. a) Find the probability that the first faulty item is the $n$th item produced, in terms of $n$.
2. b) Find the probability that the first faulty item is the fifth item or later.
3. c) Three items are selected. Find the probability that at least two are faulty.

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.

1. a) Find the probability distribution for the number of correct answers.
2. b) Calculate the probability that the student gets a pass, defined as at least 2 correct answers.
3. c) Calculate the expected number of correct answers and compare it with $3 \times \frac{1}{4}$.

Seán rolls a fair die repeatedly until he gets a 6.

1. a) Show that the probability the first 6 occurs on roll $n$ is $\left(\frac{5}{6}\right)^{n-1}\cdot\frac{1}{6}$.
2. b) Find the probability that the first 6 occurs within the first three rolls.
3. c) Find the least value of $n$ such that the probability of getting at least one 6 in $n$ rolls exceeds $0.9$.