can someone please show me the steps to solve this problem thank you 2.68 An exp
ID: 3306734 • Letter: C
Question
can someone please show me the steps to solve this problem thank you
2.68 An experiment consists of rolling a pair of (six-sided) dice and observing the sum. This experiment is repeated until the sum of 7 is observed at which point the experiment stops. Let N be the random variable which represents the number of times the experiment is repeated. That is, if the first occurrence of (sum-7) happens on the 5th roll of the dice, then N = 5. (a) Find the probability mass function for the random variable N. That is, find Py(k) = Pr(N-k) for all k. (b) What is the probability that the experiment proceeds for at least 4 rolls? That is find Pr(N 24)Explanation / Answer
The possible outcomes when a pair of die is rolled are:
Out of the above listed outcomes the favourable outcomes, that is, the possible pairs which give a sum of 7 are:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Thus, the probability of getting a favourable outcome is 6/36 =1/6
(a) It is given that the experiment is repeated until the sum 7 is observed. Thus, observing a 7 can be treated as a success. The random variable N denotes the number of trials of the experiment that have to be conducted in order to observe the first 7. Thus, the random variable N gives the trial on which the first success occurs. This is the same as a TYPE 1 GEOMETRIC DISTRIBUTION.
The possible values that the random variable N can take are 1,2,3,.......,k,.......
The probability of success in a single trial is p=1/6 and the probability of failure in a single trial is q=5/6.
Thus, in general the probaility of observing the first success in the kth trial, that is N=k is given by qk-1p
i.e k-1 trials which lead to failures and the kth trial which leads to a success, so it is q x q x q x q x....(k-1)times..x p = qk-1p .
Thus, the probability mass function of the random variable N is given as:
PN(k) = P(N=k) = qk-1p for all k
(b) The probability that the experiment proceeds for atleast 4 rolls is:
P(N4) = 1 - P(N<4)
= 1 - {P(N=1) + P(N=2) + P(N=3)}
=1-{q0p+q1p +q2p }
= 1 - {1/6 + 5/6*1/6 + 5/6*5/6*1/6}
=1 - 91/216
=125/216
1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6Related Questions
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