You draw a card from a deck of
52
cards twice, each time replacing the card. Find the following probabilities (rounded to the nearest thousandths) of observing:
a.) exactly 1 black card.
P(1 black card)=
b.) exactly 1 club. P(1 club)=
c.) exactly 1 two. P(1 two)
a) not black ---> white
so you could have WB or BW
prob(WB or BW) = (1/2)(1/2) + (1/2)(1/2) = 1/2
or, think of it this way:
outcomes are WW, WB, BW, BB for a total of 4 possibilities, both W and B have equal likely outcomes
two of those are the outcomes you want ---> 2/4 = 1/2
b.) exactly 1 club. , let X be non-club
prob(club) = 13/52 = 1/4
prob(X) = 3/4
you want Prob(CX or XC) = (1/4)(3/4) + (3/4)(1/4) = 3/8
do c) following the method of b)
To find the probabilities, we need to consider the total number of possible outcomes and the number of favorable outcomes.
a.) To find the probability of observing exactly 1 black card, we first need to determine the number of black cards in a deck. A standard deck of 52 cards contains 26 black cards (clubs and spades). Since we are replacing the cards after drawing, each draw is an independent event.
The probability of drawing a black card on the first draw is 26/52 = 1/2.
The probability of not drawing a black card on the first draw is 1 - 1/2 = 1/2.
The probability of drawing a black card on the second draw is also 1/2.
To calculate the probability of exactly 1 black card, we need to consider two cases:
1. Drawing a black card on the first draw and not drawing one on the second.
2. Not drawing a black card on the first draw and drawing one on the second.
P(1 black card) = (1/2) * (1/2) + (1/2) * (1/2)
= 1/4 + 1/4
= 1/2
Therefore, the probability of observing exactly 1 black card is 1/2.
b.) To find the probability of observing exactly 1 club, we need to determine the number of club cards in a deck. A standard deck of 52 cards contains 13 club cards. Since we are replacing the cards after drawing, each draw is an independent event.
The probability of drawing a club card on the first draw is 13/52 = 1/4.
The probability of not drawing a club card on the first draw is 1 - 1/4 = 3/4.
The probability of drawing a club card on the second draw is also 1/4.
To calculate the probability of exactly 1 club, we need to consider two cases:
1. Drawing a club card on the first draw and not drawing one on the second.
2. Not drawing a club card on the first draw and drawing one on the second.
P(1 club) = (1/4) * (3/4) + (3/4) * (1/4)
= 3/16 + 3/16
= 6/16
= 3/8
Therefore, the probability of observing exactly 1 club is 3/8.
c.) To find the probability of observing exactly 1 two, we need to determine the number of twos in a deck. A standard deck of 52 cards contains 4 twos. Since we are replacing the cards after drawing, each draw is an independent event.
The probability of drawing a two on the first draw is 4/52 = 1/13.
The probability of not drawing a two on the first draw is 1 - 1/13 = 12/13.
The probability of drawing a two on the second draw is also 1/13.
To calculate the probability of exactly 1 two, we need to consider two cases:
1. Drawing a two on the first draw and not drawing one on the second.
2. Not drawing a two on the first draw and drawing one on the second.
P(1 two) = (1/13) * (12/13) + (12/13) * (1/13)
= 12/169 + 12/169
= 24/169
Therefore, the probability of observing exactly 1 two is 24/169.
To find the probabilities, we need to determine the number of favorable outcomes and the total number of possible outcomes.
a.) To find the probability of getting exactly 1 black card, we need to calculate the number of ways we can get 1 black card from 2 draws, while taking into account the replacement of the card.
Number of favorable outcomes:
In the first draw, there are 26 black cards and 26 non-black cards. So, the probability of drawing a black card in the first draw is 26/52 = 1/2.
In the second draw, there is still an equal probability of drawing a black card. Hence, the probability of not drawing a black card in the second draw is 1/2.
Since the probability of drawing exactly 1 black card can occur in two ways (black-non-black or non-black-black), we need to calculate the probability for each case.
Probability of black-non-black: (1/2) * (1/2) = 1/4
Probability of non-black-black: (1/2) * (1/2) = 1/4
Total number of favorable outcomes: 1/4 + 1/4 = 1/2
Total number of possible outcomes: 2 (since there are two draws)
P(1 black card) = favorable outcomes / total outcomes = 1/2 / 2 = 1/4 = 0.250 (rounded to the nearest thousandths)
Therefore, the probability of observing exactly 1 black card is 0.250.
b.) To find the probability of getting exactly 1 club, we use a similar approach as above.
Number of favorable outcomes:
In each draw, there are 13 clubs and 39 non-club cards. So, the probability of drawing a club card in the first draw is 13/52 = 1/4, and the probability of not drawing a club card is 3/4.
The same probabilities apply to the second draw.
Probability of club-non-club: (1/4) * (3/4) = 3/16
Probability of non-club-club: (3/4) * (1/4) = 3/16
Total number of favorable outcomes: 3/16 + 3/16 = 3/8
Total number of possible outcomes: 2 (since there are two draws)
P(1 club) = favorable outcomes / total outcomes = 3/8 / 2 = 3/16 = 0.188 (rounded to the nearest thousandths)
Therefore, the probability of observing exactly 1 club is 0.188.
c.) To find the probability of getting exactly 1 two, we need to consider that there are only 4 twos in the deck.
Number of favorable outcomes:
In each draw, there are 4 twos and 48 non-two cards. So, the probability of drawing a two in the first draw is 4/52 = 1/13, and the probability of not drawing a two is 12/13.
The same probabilities apply to the second draw.
Probability of two-non-two: (1/13) * (12/13) = 12/169
Probability of non-two-two: (12/13) * (1/13) = 12/169
Total number of favorable outcomes: 12/169 + 12/169 = 24/169
Total number of possible outcomes: 2 (since there are two draws)
P(1 two) = favorable outcomes / total outcomes = 24/169 / 2 = 24/338 = 0.071 (rounded to the nearest thousandths)
Therefore, the probability of observing exactly 1 two is 0.071.