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# The Binomial Distribution

This tutorial covers material encountered in chapter 14 of the VCE Mathematical Methods Textbook, namely:

• Bernoulli distribution
• Binomial distribution
• Mean and variance of a binomial distribution

## Worksheet

Q1. If X is a binomial distribution with parameters n=5 and p=\dfrac{1}{4}, find:

(a) P(X=0)

(b) P(X=1)

(c) P(X=2)

(d) P(X\leq 1)

(e) P(X>2)

Q2. Arshar, who plays netball, knows he has a probability of 0.75 to score a point when he goes for the goal. What’s the probability that if he tries to go for the goal four times during a game he will score on exactly three of those shots?

Q3. Jeremy’s chocolate machine has a probability of 0.2 of making a defective chocolate bar.

(a) What’s the expected number of good (non-defective) chocolate bars in a day if Jeremy’s machine produces 100 chocolate bars every day?

(b) What is the standard deviation of the number of defective chocolates?

Q4. A fair, standard six sided die is cast ten times, the probability of getting an even number exactly four times is a\times\left(\dfrac{1}{2}\right)^{10},\,a\in\N. Find a.

Q5. When playing Supa Smush Bros. Melee, Tasman has a 30% chance of beating his friend Nathan every game. How many games does Tasman and Nathan need to play in order for Tasman to have a 0.95 probability of winning at least one game?

Q6. A binomial distribution has mean \mu =3 and standard deviation \sigma=\dfrac{3}{2}. Find the p, the probability of success in a single trail.

Q7. Given a fair, standard six sided die what’s the probability of rolling 4 or under seven times within ten rolls, given that the first roll was a 6?

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