The binomial distribution Problem set A

For each situation, decide whether X can be modelled by a binomial distribution. If it can, write X ~ B(n, p). If it cannot, say which condition fails.

1. (a) A fair eight-sided die is rolled 24 times. X is the number of 5s.
2. (b) Cards are drawn one at a time from a pack, each card being put back and the pack shuffled. X is the number of cards drawn until the first heart appears.
3. (c) A box holds 9 chocolates, 4 of them dark. Three chocolates are chosen at random and eaten. X is the number of dark chocolates chosen.
4. (d) In a large production run, 5% of tablets are faulty. 30 tablets are picked at random. X is the number that are faulty.

\(X \sim B(16, 0.4)\). Find

1. (a) \(P(X = 6)\)
2. (b) \(P(X \le 5)\)
3. (c) \(P(X > 9)\)
4. (d) \(P(3 < X \le 7)\)

\(X \sim B(30, 0.12)\). Find

1. (a) \(P(X < 4)\)
2. (b) \(P(X \ge 2)\)

In a large delivery of mugs, 7% are chipped. A restaurant picks 20 mugs at random. X is the number of chipped mugs in the sample.

1. (a) Write down the distribution of X.
2. (b) Find the probability that exactly 2 mugs in the sample are chipped.
3. (c) Find the probability that at least 3 mugs in the sample are chipped.
4. (d) Find the probability that between 1 and 4 mugs in the sample are chipped, inclusive.

A bag holds 7 red sweets and 5 yellow sweets. Four sweets are taken out at random, one after another, without being put back. X is the number of red sweets taken.

1. (a) Explain why X is not binomial.
2. (b) Find the probability that the second sweet is red, given that the first sweet was red. Compare it with the probability that the first sweet is red.

Each visitor to a website independently signs up with probability 0.3. 25 visitors arrive. X is the number who sign up.

1. (a) Find the probability that at least 5 but fewer than 10 visitors sign up.
2. (b) Find the probability that at least 10 visitors sign up.

20% of the trees in a forest have a fungus. The fungus spreads from a tree to its neighbours. 15 trees growing next to each other are checked. X is the number with the fungus. Explain why X may not be binomial.