find the probability that a randomly chosen offspring survives to maturity.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. The offspring of an animal can be split into three distinct types: type BB, type Bb and type bb. If the offspring is of type bb it has brown fur, otherwise it has black fur. Suppose that the initial proportion of type BB offspring is 1/8, the initial proportion of type bb offspring is 1/4, and the rest of the offspring are of type Bb. Suppose that any offspring of type BB, type Bb and type bb will independently survive to maturity with probability 1/4, 1/8 and 3/4 respectively.

(a)  Find the probability that a randomly chosen offspring survives to maturity.

(b)  Given that an offspring has survived to maturity, find the probability that it has black fur.

2. We toss a coin which shows a head with probability p. If the coin shows a head, we toss it again, but if it shows a tail we stop (so the coin is tossed either once or twice).

(a)  Find the probability that we get a head at the end of this experiment.

(b) Now we toss 10 coins, where each coin shows a head with probability p, independently of each other. Each coin which shows a head is tossed again. What is the probability mass function of the number of heads we have after the second round of tosses?

(c)  Find the expected number of heads after the first round of tosses in Q2(b). How does this number change after the second round of tosses?

(d) R: Use simulation to find the probability that we have no heads at the end of the experiment in Q2(b) when p = 0.3

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