devin and Tej are playing card. Devin draws a card, replaces it and then shuffles the pack. Tej then draws a card. find the probability of
a)both draw an ace
b) both draw the king of hearts
c) Devin draws a spade and Tej draws a queen
d)exactly one of the cards draws is a heart
e)both cards are red or both cards are black
f)the cards are different in colors
Answers
Step-by-step explanation:
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a) Both draw an ace = 1/169
b) Both draw the king of hearts = 1/2704
c) Devin draws a spade and Tej draws a queen = 1/52
d) Exactly one of the card draws is a heart = 0.44
e) Both cards are red or both cards are black = 0.490
f) The cards are different in colors = 0.94
Given:
Devin and Tej are playing cards. Devin draws a card, replaces it, and then shuffles the pack. Tej then draws a card.
To find:
Find the probability of
a) Both draw an ace
b) Both draw the king of hearts
c) Devin draws a spade and Tej draws a queen
d) Exactly one of the card draws is a heart
e) Both cards are red or both cards are black
f) The cards are different in colors
Solution:
From the data,
Devin and Tej are playing cards.
There are 52 cards in a standard deck
Number of aces = 4
Number of king hearts = 4
Number of spades = 13
Number of red cards = 26
Number of black cards = 26
Assuming the deck is well-shuffled, the probability of each event can be calculated as follows:
a) Both drawing an ace
The probability that Devin draws an ace = 4/52 = 1/13.
The probability that Tej then draws an ace = 1/13
P(both draw an ace) = (1/13) x (1/13) = 1/169
b) Both draw the king of hearts
The probability that Devin draws the king of hearts = 1/52
The probability that Tej then draws the king of hearts = 1/52
P(both draw the king of hearts) = (1/52) x (1/52) = 1/2704
c) Devin draws a spade and Tej draws a queen
The probability that Devin draws a spade = 13/52 = 1/4
The probability that Tej then draws a queen = 4/52 = 1/13
P(Devin draws a spade and Tej draws a queen) = (1/4) x (1/13) = 1/52
d) Exactly one of the card draws is a heart
The probability of drawing 2 kings of hearts = 1/2704
The probability that none of the cards are heart = 39/52 × 38/52
P(neither card is a heart) = (39/52) x (38/51) ≈ 0.559
Therefore, the probability that at least one card is a heart is:
P(at least one card is a heart) = 1 - P(neither card is a heart) = 0.45
P(exactly one card is a heart)
= P(at least one card is a heart) - P(both cards are hearts)
= 0.45 - 1/2704 = 0.44
e) Both cards are red or both cards are black
The probability that both cards are red = (26/52) × (25/51)
[since Devin replaced the first card drawn]
The probability that both cards are black = (26/52) × (25/51).
Therefore,
P(both cards are red or both cards are black)
= (26/52) x (25/51) + (26/52) x (25/51) = 25/51 = 0.490
f) The cards are different in colors
The probability that both cards are red = (26/52) x (25/51) = 0.24
The probability that both cards are black = (26/52) x (25/51) = 0.24
The probability that both are same color = (0.24)(0.24) = 0.0576
Therefore,
The probability that both are different colors = 1 - 0.0576 = 0.94
Therefore,
a) Both draw an ace = 1/169
b) Both draw the king of hearts = 1/2704
c) Devin draws a spade and Tej draws a queen = 1/52
d) Exactly one of the card draws is a heart = 0.44
e) Both cards are red or both cards are black = 0.490
f) The cards are different in colors = 0.94
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