clarkkent0610 wrote:

For each players turn in a certain board game, a card is drawn. 3/4 of the cards in the deck are marked with a circle, and the remaining cards are marked with a square. If five players draw a card and then return it to the deck, what is the probability that at least four of the cards drawn are marked with a square?

A. 1/4^3

B. 1/4^4

C. 5*(1/4^3)

D. 1/4^5

E. (3/2)*(1/4^4)

The probability of circle is 3/4 and the probability of square is 1-3/4=1/4.

We want the probability that out of 5 cards drawn 4 OR 5 are squares.

The probability of 4 squares or the probability of SSSSC, is \(P(SSSSC)=\frac{5!}{4!}*(\frac{1}{4})^4*(\frac{3}{4})=\frac{15}{4^5}\). We are multiplying by \(\frac{5!}{4!}=5\) since SSSSC scenario can occur in 5 ways: SSSSC, SSSCS, SSCSS, SCSSS, and CSSSS (number of premutations of 5 letters SSSSC out of which 4 S's are identcal);

The probability of 5 squares or the probability of SSSSS, is simply \(P(SSSSS)=(\frac{1}{4})^5\).

Therefore the overall probability is \(\frac{15}{4^5}+\frac{1}{4^5}=\frac{1}{4^3}\).

Answer: A.

P.S. Please indicate OA for PS problems.

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