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...

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

Extracted 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

Jun 02, 2022

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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...

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