Answer:
The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement is 0.03125 .
The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement is 0.03058
Step-by-step explanation:
Total no. of cards = 52
Spade cards = 13 (Black)
Club cards = 13 (Black)
Heart cards = 13 (Red)
Diamond cards = 13(Red)
Total red cards = 26
With replacement case:
Probability of getting spade on first draw = ![\frac{13}{52}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B52%7D)
Now card is replaced
Probability of getting spade on second draw = ![\frac{13}{52}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B52%7D)
Now card is replaced
Probability of getting red card on third draw = ![\frac{26}{52}](https://tex.z-dn.net/?f=%5Cfrac%7B26%7D%7B52%7D)
So, The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement = ![\frac{13}{52} \times \frac{13}{52} \times \frac{26}{52}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B52%7D%20%5Ctimes%20%5Cfrac%7B13%7D%7B52%7D%20%5Ctimes%20%5Cfrac%7B26%7D%7B52%7D)
=![\frac{1}{32}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B32%7D)
= ![0.03125](https://tex.z-dn.net/?f=0.03125)
Without replacement case:
Probability of getting spade on first draw = ![\frac{13}{52}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B52%7D)
Remaining cards = 51
Remaining spade cards = 12
Probability of getting spade on second draw = ![\frac{12}{51}](https://tex.z-dn.net/?f=%5Cfrac%7B12%7D%7B51%7D)
Remaining cards = 50
Probability of getting red card on third draw = ![\frac{26}{50}](https://tex.z-dn.net/?f=%5Cfrac%7B26%7D%7B50%7D)
So, The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement = ![\frac{13}{52} \times \frac{12}{51} \times \frac{26}{50}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B52%7D%20%5Ctimes%20%5Cfrac%7B12%7D%7B51%7D%20%5Ctimes%20%5Cfrac%7B26%7D%7B50%7D)
= ![\frac{13}{425}](https://tex.z-dn.net/?f=%5Cfrac%7B13%7D%7B425%7D)
= ![0.03058](https://tex.z-dn.net/?f=0.03058)
Hence The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement is 0.03125 . The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement is 0.03058