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Chapter 4: Q.4.25 (page 165)
Two coins are to be 铿俰pped. The 铿乺st coin will land on heads with probability ., the second with probability .. Assume that the results of the 铿俰ps are independent, and let X equal the total number of heads that result. (a) Find P{X =}. (b) Determine E[X].
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A student is getting ready to take an important oral examination and is concerned about the possibility of having an 鈥渙n鈥� day or an 鈥渙ff鈥� day. He figures that if he has an on the day, then each of his examiners will pass him, independently of one another, with probability, whereas if he has an off day, this probability will be reduced to. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on the day, should he request an examination withexaminers or withexaminers?
Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students who were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on her bus.
(a) Which of E[X] or E[Y] do you think is larger? Why?
(b) Compute E[X] and E[Y].
There are N distinct types of coupons, and each time one is obtained it will, independently of past choices, be of type i with probability Pi, i = 1, ... , N. Let T denote the number one need select to obtain at least one of each type. Compute P{T = n}.
When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?
A fair coin is flipped times. Find the probability that there is a string of consecutive heads by
(a) using the formula derived in the text;
(b) using the recursive equations derived in the text.
(c) Compare your answer with that given by the Poisson approximation.
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