Please Explain! An urn contains seven chips labeled 1,2,...,7. Three of the chips are black, two...

Please Explain!

An urn contains seven chips labeled 1,2,...,7. Three of the chips are black, two are red, and two are green. The chips are drawn randomly one at a time without replacement until the urn is empty. Answer both questions for i = 1,...,7.

a. What is the probability that ith draw is chip 5?

b. What is the probability that ith draw is black?

Solutions

Expert Solution

a. There are 7 numbers which have equal probability of being drawn.

Thus the probability that the first chip is chip 5 is 1/7.

If the first chip is not chip 5 (whose probability is 6/7), the probability that the second chip is chip 5 is 1/6. Thus the probability that the second chip is chip 5 is again 1/7 and so on.

The probability that the ith draw is chip 5 is therefore 1/7.

b. There are three black and four non black chips.

The probability that the first draw is black is 3/7.

If the first draw is black, the probability that the second is also black is 2/6 while if the first draw is not black, the probability that the second is black is 3/6. Thus the probability that the second draw is black = 3/7 * 2/6 + 4/7 * 3/6 = 18/42 = 3/7.

Similarly the probability that the third draw is black is 3/7 * 2/6 * 1/5 + 3/7 * 4/6 * 2/5 + 4/7 * 3/6 * 2/5 + 4/7 * 3/6 * 3/5 = 90/210 = 3/7 and so on.

Therefore, the probability that the ith draw is black is 3/7.

orchestra answered 2 years ago

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