BTGmoderatorLU wrote:Three men and 2 women will present 5 consecutive speeches, 1 by each person, at a conference. If the order of the speakers is determined randomly, what is the probability that at least 2 of the men's speeches will be consecutive?
A. 3124/3125
B. 9/10
C. 4/5
D. 16/25
E. 1/2
P(at least 2 men speak consecutively) = 1 - P(no men speak consecutively).
For no men to speak consecutively, the order of the speeches must be as follows:
man-woman-man-woman-man.
P(1st speaker is a man) = 3/5. (Of the 5 speakers, 3 are men.)
P(2nd speaker is a woman) = 2/4. (Of the 4 remaining speakers, 2 are women.)
P(3rd speaker is a man) = 2/3. (Of the 3 remaining speakers, 2 are men.)
P(4th speaker is a woman) = 1/2. (Of the 2 remaining speakers, 1 is a woman.)
P(5th speaker is a man) = 1/1. (Only 1 speaker remains, and he is a man.)
To combine the probabilities above, we multiply:
P(no men speak consecutively) = 3/5 * 2/4 * 2/3 * 1/2 * 1/1 = 1/10.
Thus:
P(at least 2 men speak consecutively) = 1 - 1/10 = 9/10.
The correct answer is
B.
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