Description
Exercise 6.1. This question involves independent coin tosses. For parts (a) and (b), give
the answer in two ways: as an algebraic expression in terms of quantities p, n, and k, and
as a number. (Use Matlab to do the calculations.)
(a) A fair two-sided coin is tossed 5 times. What is the probability that exactly three of
these tosses produce heads? Explain your answer.
(b) A fair two-sided coin is tossed 40 times. What is the probability that exactly 20 of
these tosses will produce heads? Explain your answer and specify how you calculated
the associated number.
(c) Under the same circumstances as in part (b), explain how you could justify saying
“The most likely outcome of the 40 tosses is 20 heads”, including a clear definition of
“most likely”.
Exercise 6.2. This problem refers to the days when people wore hats and checked them
when going to a restaurant.
Suppose that n people come to the restaurant in a group and check their hats. Unfortunately, the person in charge of the hat check is totally disorganized and does not keep
track of which hat belongs to each person. Thus, when the people leave the restaurant, each
person’s probability of getting his/her own hat is 1/n. Let Ai be the event that person i
gets his or her own hat when leaving the restaurant; let the random variable Xi be equal to
1 if person i gets his/her own hat back, and Xi = 0 otherwise. Assume that n = 4.
(1) What is the probability of A1? What is the expected value of X1? Explain your answer.
(2) What is the probability of the event A1∩A2, i.e., that both persons 1 and 2 will receive
their own hats? What is the expected value of the random variable X1 × X2? Explain
your answer.
(3) In general, are events Ai and Aj
, where i 6= j, independent? Explain your answer.
Exercise 6.3. Assume that it rains in a big city on half of the days. The weather forecaster
is reasonably reliable.
• Given a forecast of rain, the probability is 2/3 that it will rain.
2
• Given a forecast that it will not rain, the probability is 2/3 that it will not rain.
A professor of probability relies to some extent on the forecasts, but is exceptionally cautious
about rain: when rain is forecast, the professor brings an umbrella to the office; when the
forecast is that it will not rain, the professor brings an umbrella with probability 1/3.
(1) The probability that the professor does not bring an umbrella to the office, given that
it rains that day, is 2/9. Show how this correct answer is obtained, giving all relevant
formulas used to calculate probabilities and explaining each step in your calculation.
(2) The probability that the professor brings an umbrella to the office, given that it does
not rain that day, is 5/9. Show how this correct answer is obtained, giving all relevant
formulas used and explaining each step of your calculation.
Exercise 6.4. A restaurant offers two kinds of pie (strawberry and cherry), and always
begins each day with an equal number of pies of these two kinds. Every day exactly 10
customers each request a pie, and the probability that a given customer will choose one kind
or the other is 1/2.
(a) If the restaurant stocks 5 strawberry pies and 5 cherry pies every day, what is the
probability that every customer will receive his/her requested kind of pie? Explain
your answer.
(b) Answer the same question as in part (a), but assuming that the restaurant stocks 8
strawberry pies and 8 cherry pies each day. Explain your answer.
Exercise 6.5. Given a fair coin, a gambler will win a large amount of money if the gambler
manages to toss “heads”, and the gambler has three chances to do so. Once the coin comes
up “heads”, the tosses will stop. Otherwise, the gambler may make three tosses in total. Let
X be the number of times that heads is tossed, and Y be the number of times that tails is
tossed. For each case, explain how you obtained your results.
(1) For X, give (i) the expected value and (ii) the variance.
(2) For Y , give (i) the expected value and (ii) the variance.
