a card is picked from a pack containing 52 cards . it is then replaced and a second card is picked. Find the probability that
a)both cards are the seven of diamonds,
b)the first card is a heart and the second a spade.
c)one card is from a black suit and other is from a
red suit
d) at least one card is a queen
I will do two:
b) Pr= 13/52 * 13/52
d)Pr(one a queen)= pr(two queens)+pr(one queen)
Pr(QQ)+Pr(QN)+Pr(NQ)
4/52*4/52 + 4/52*48/52 + 4/52*48/52
= (16+2*4*48)/52*52=.148 check it.
To find the probability in each of these cases, we need to determine the total number of possible outcomes and the number of favorable outcomes.
a) Probability of both cards being the seven of diamonds:
There is only 1 seven of diamonds in a pack of 52 cards. Since the first card is replaced before picking the second one, the probability of getting a seven of diamonds twice is:
Number of favorable outcomes: 1
Total number of outcomes: 52 * 52 (since there are 52 possible outcomes for each card)
Probability = (1/52 * 1/52) = 1/2704
b) Probability of the first card being a heart and the second card being a spade:
There are 13 hearts and 13 spades in a deck of cards. Since the first card is replaced before picking the second one, the probability of getting a heart followed by a spade is:
Number of favorable outcomes: 13 * 13 (one heart followed by one spade)
Total number of outcomes: 52 * 52
Probability = (13/52 * 13/52) = 1/16
c) Probability of one card being from a black suit and the other card from a red suit:
There are 26 black cards (clubs and spades) and 26 red cards (hearts and diamonds) in a deck of cards. The probability of getting one card from a black suit and the other card from a red suit is:
Number of favorable outcomes: 26 * 26 (one black card followed by one red card)
Total number of outcomes: 52 * 52
Probability = (26/52 * 26/52) = 1/4
d) Probability of at least one card being a queen:
The probability of at least one card being a queen can be calculated by finding the probability of the complement event (no queen) and subtracting it from 1. There are 4 queens in a deck of cards. The probability of no queen is:
Number of favorable outcomes: 48 * 48 (since there are 48 possible outcomes for each non-queen card)
Total number of outcomes: 52 * 52
Probability of no queen = (48/52 * 48/52) = 36/49
Probability of at least one queen = 1 - Probability of no queen = 1 - (36/49) = 13/49
So, the probabilities are:
a) 1/2704
b) 1/16
c) 1/4
d) 13/49