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INDEPENDENCE OF EVENTS 37 10. Show that the converse of Theorem 2 also holds. Thus A and B are independent if, and only if, A and Be are independent, and so on 11, A lot of five identical batteries is life tested. The probability assignment is assumed to be P(A)= (1/A)e/Adx for any event AC 0,00), where A > 0 is a known constant. Thus the probability that a battery fails after time t is given by P(t, 00)= (1/A)e/dx, t0. If the times to failure of the batteries are independent, what is the probability that at least one battery will be operating after to hours? 12. On =(a, b), -oo

INDEPENDENCE OF EVENTS 37 10. Show that the converse of Theorem 2 also holds. Thus A and B are independent if, and only if, A and Be are independent, and so on 11, A lot of five identical batteries is life tested. The probability assignment is assumed to be P(A)= (1/A)e/Adx for any event AC 0,00), where A > 0 is a known constant. Thus the probability that a battery fails after time t is given by P(t, 00)= (1/A)e/dx, t0. If the times to failure of the batteries are independent, what is the probability that at least one battery will be operating after to hours? 12. On =(a, b), -oo

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 56E
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INDEPENDENCE OF EVENTS 37 10. Show that the converse of Theorem 2 also holds. Thus A and B are independent if, and only if, A and Be are independent, and so on 11, A lot of five identical batteries is life tested. The probability assignment is assumed to be P(A)= (1/A)e/Adx for any event AC 0,00), where A > 0 is a known constant. Thus the probability that a battery fails after time t is given by P(t, 00)= (1/A)e/dx, t0. If the times to failure of the batteries are independent, what is the probability that at least one battery will be operating after to hours? 12. On =(a, b), -oo<a< b <o0, each subinterval is assigned a probability propor- tional to the length of the interval. Find a necessary and sufficient condition for two events to be independent. 13. A game of craps is played with a pair of fair dice as follows. A player rolls the dice. If a sum of 7 or 11 shows up, the player wins; if a sum of 2, 3, or 12 shows up, the player loses. Otherwise the player continues to roll the pair of dice until the sum is either 7 or the first number rolled. In the former case the player loses and in the latter the player wins. (a) Find the probability that the player wins on the nth roll. (b) Find the probability that the player wins the game. (c) What is the probability that the game ends on: (i) the first roll, (ii) second roll, and (iii) third roll?
Transcribed Image Text:INDEPENDENCE OF EVENTS 37 10. Show that the converse of Theorem 2 also holds. Thus A and B are independent if, and only if, A and Be are independent, and so on 11, A lot of five identical batteries is life tested. The probability assignment is assumed to be P(A)= (1/A)e/Adx for any event AC 0,00), where A > 0 is a known constant. Thus the probability that a battery fails after time t is given by P(t, 00)= (1/A)e/dx, t0. If the times to failure of the batteries are independent, what is the probability that at least one battery will be operating after to hours? 12. On =(a, b), -oo<a< b <o0, each subinterval is assigned a probability propor- tional to the length of the interval. Find a necessary and sufficient condition for two events to be independent. 13. A game of craps is played with a pair of fair dice as follows. A player rolls the dice. If a sum of 7 or 11 shows up, the player wins; if a sum of 2, 3, or 12 shows up, the player loses. Otherwise the player continues to roll the pair of dice until the sum is either 7 or the first number rolled. In the former case the player loses and in the latter the player wins. (a) Find the probability that the player wins on the nth roll. (b) Find the probability that the player wins the game. (c) What is the probability that the game ends on: (i) the first roll, (ii) second roll, and (iii) third roll?
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