Math 135, Fall 2011: HW 6
This homework is due at the beginning of class on
Wednesday October 19th, 2011
. You are
free to talk with each other and get help.
However, you should write up your own solutions and
understand everything that you write.
Problem 1(p.217 #1).A coin which lands heads with probabilitypis tossed repeatedly. Assumingindependence of the tosses, find formulae for
a) the probability that exactly 5 heads appear in the first 9 tosses;b) the probability that the first head appears on the 7th toss;c) the probability that the fifth head appears on the 12th toss;d) the probability that the same number of heads appear in the first 8 tosses as in the next 5tosses.Part (a) is a binomial (n,p) probability, wheren= 9. Letq= 1-p. We getP(exactly 5 heads in first 9 tosses) =95p5q4.Part (b) is a geometric distribution; in order for the first head to appear at the 7th toss, the previous6 tosses must be tails and the 7th toss must be heads. By independence, this isq6p.Part (c) is a negative binomial distribution. In order for the fifth head to appear on the 12th toss,the 12th toss must be a heads, and there must be exactly 4 heads in the previous 11 tosses. ThusP(5th head appears on 12th toss) =114p4q7p=114p5q7.For part (d), observe that by independence of trials, the random variablesX1= [number of heads in the first 8 tosses]X2= [number of heads in the next 5 tosses]are independent. AlsoX1∼B(8, p), andX2∼B(5, p). HenceP(same # of H in first 8 as in next 5 tosses) =5Xk=1P(X1=k, X2=k)=5Xk=1P(X1=k)P(X2=k)=5Xk=18kpkq8-k5kpkq5-kProblem 2(p.218 #4).A game is played in which three people each toss a fair coin to see if oneof their coins shows a different face from the other two. The game ends if one of the three peoplehas a coin which lands differently from the other two.
1

a) What is the probability of the game ending after one round of play?b) What is the probability that the game lastsrrounds?c) What is the expected duration of play?

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