A standard deck of playing cards is shuffled and three people each choose a card. Find the probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement, and if the cards are chosen without replacement. (a) The cards are chosen with replacement. (b) The cards are chosen without replacement.

Answer:Part (a) The cards are chosen with replacement is 0.03125 or 1/32Part (b) The cards are chosen without replacement is 0.0306 or 13/425.Step-by-step explanation:Consider the provided information.A standard deck of playing cards is shuffled 3 cards are chosen.In a deck of card 13 are spade out of 52 cards. 26 cards are red out of 52 cards.We need to find the probability that the first two cards chosen are spades and the third card is red.Part (a) The cards are chosen with replacement.Suppose the first card is spade then the probability is P(A) = 13/52As spade are 13 out of 52.Now, the card is replaced back in the deck and we again draw a card. The probability of getting spade in second time is P(B) = 13/52.The card is replaced back in the deck and we again draw a card. The probability of getting a red card in third time is P(C) = 26/52. Now Multiply 3 fraction together as shown.[tex]P(A)\times P(B) \times P(C) =\frac{13}{52} \times \frac{13}{52} \times \frac{26}{52}\\P(A)\times P(B) \times P(C) =\frac{1}{4}\times \frac{1}{4}\times \frac{1}{2}\\P(A)\times P(B) \times P(C) =\frac{1}{32}=0.03125[/tex]Hence, the probability of getting the first two cards chosen are spades and the third card is red if the cards are chosen with replacement is 0.03125 or 1/32.Part (b) The cards are chosen without replacement.Suppose the first card is spade then the probability is P(A) = 13/52Now, the card is not replaced back in the deck and we again draw a card. The probability of getting spade in second time is P(B) = 12/51.The probability of getting a red card in third time is P(C) = 26/50. Now Multiply 3 fraction together as shown.[tex]P(A)\times P(B) \times P(C) =\frac{13}{52} \times \frac{12}{51} \times \frac{26}{50}\\P(A)\times P(B) \times P(C) =\frac{13}{425}=0.0306[/tex]Hence, the probability of getting the first tow cards chosen are spades and the third card is red if the cards are chosen without replacement is 0.0306 or 13/425.