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A coin flip game at a carnival consists of 2 rounds. The probability that a player wins the first round is

1 / 4

, and the probability that the player wins the second round is

1 / 8

. Assuming the outcome of each round is independent of the other, what is the probability that a player will win the first round and lose the second round?

(Note: Each round must be either won or lost.)

		A. 7 / 16
		B. 9 / 16
		C. 1 / 32
		D. 7 / 32

Hint :
To find the probability that two independent events occur, multiply together their individual probabilities.

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Explanation

It is given that the outcome of each round is independent of the other, so winning the first round and losing the second round are independent events .

To find the probability that two independent events occur, multiply together their individual probabilities.

The question asks for the probability that a player wins the first round and loses the second round, but the given probabilities are that the player wins each round.

1st round		2nd round
1 / 4	P(win)	1 / 8

The event that a player loses a round is the complement of the event that the player wins the round.

The probability of the complement of an event is 1 minus the probability of the event. Therefore, the probability that a player loses a round is 1 minus the probability that the player wins the round.

To find the probability that a player loses the second round, plug the probability that the player wins the second round (

1 / 8

) into the complement probability formula.

P(lose 2nd) = 1 − P(win 2nd)	Complement formula
P(lose 2nd) = 1 − 1 / 8	Plug in P(win 2nd)
P(lose 2nd) = 7 / 8	Subtract: 1 − 1 / 8 = 8 / 8 − 7 / 8

The probability that the player will win the first round is
1 / 4
, and the probability that the player will lose the second round is
7 / 8
.

Multiply these probabilities together to find the probability that the player will win the first round and lose the second round.

1 / 4 · 7 / 8	P(win 1st) · P(lose 2nd)
7 / 32	Multiply

The probability that a player will win the first round and lose the second round is

7 / 32

(Choice A)

7 / 32

may result from a combination of mistakenly finding the probabilities that the player will lose the first round and win the second (instead of win the first and lose the second) and the error described in Choice B.

(Choice B)

9 / 16

may result from mistakenly calculating the average of the probabilities that the player will win the first round and lose the second round, but it is necessary to multiply the probabilities of individual independent events to find the probability that they both occur.

(Choice C)

1 / 32

is the probability that the player will win both rounds, but the question asks for the probability that the player will win the first round and lose the second round.

(Choice D)

3 / 32

is the probability that the player will lose the first round and win the second round, but the question asks for the probability that the player will win the first round and lose the second round.

Things to remember:

To find the probability that two independent events occur, multiply together their individual probabilities.
The complement of an event A is the event that A does NOT occur. The probability that an event A does NOT occur, P(not A), is equal to 1 minus the probability that it does occur, P(A):

P(not A) = 1 − P(A)

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