Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told. a. ?What is the probability that the rumor is repeated in the order B, C, D, E, and F? b. ?What is the probability that F is the third person at the party to be told the rumor? c.? hat is the probability that F is the last person to hear the rumor? d. ?If at each stage the person who currently “has” the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the party?

Answer: Step 1 of 1 Given, Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told. a. The number of possible ways to select the persons in the order B, C, D, E and F is 5 × 4 × 3 × 2 × 1 = 120 1 Probability that the rumour is repeated in order B, C, D, E and F is 120 = 0.0083 b. The number of possible ways to select the person F as third is 4 × 3 × 1 × 2 × 1= 24 24 The probability that F is the third person at the party to be told the rumor is120 = 0.2 c. The number of possible ways that F is the last person is 4 × 3 × 2 × 1 × 1 = 24 The probability that F is the last person to hear the rumor is 120 = 0.2. d. If at each stage the person who currently “has” the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, P(F has not heard after 10 times) = P(Not on #1 Not on #2 . . . Not on #10) 4 4 5 × . . . × 5 4 10 = ( )5 = 0.1073 Therefore, the probability that F has still not heard the rumor after it has been told ten times at the party is 0.1073.