The Binomial Distribution [Video Tutorial]

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

Q1 – Find Probabilities of Binomial Distributions

Q2 – Application of Binomial Distribution – Netball Shooting

Q3 – Application of Binomial Distribution – Chocolate Manufacturing

Q4 – Application of Binomial Distribution – Rolling Dice

Q5 – Application of Binomial Distribution – Playing Games

Q6 – Mean and Variance of a Binomial Distribution

Q7 – Application of Binomial Distribution – Rolling Dice 2

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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