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Question

In a series of 3 independent trials, the probability of exactly 2 success is 12 times as large as the probability of 3 successes. The probability of a success in each trail is


A
15
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B
25
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C
35
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D
45
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Solution

The correct option is A $$\dfrac{1}{5}$$
For a binomial experiment consisting of n trials, the probability of exactly
k successes is
$$P(k \text{ successes}) = ^nC_kp^k(1-p)^{n-k}$$
where the probability of success on each trial is p.
According to the question,
$$n=3,, P(\text{2 successes})=12\times P(\text{3 successes})$$
To find:
The probability of a success in each trail, p
$$P(\text{2 successes})=12\times P(\text{3 successes})\\\implies ^3C_2p^2(1-p)^{3-2}=12 \times ^3C_3p^3(1-p)^{3-3}\\\implies 3\times p^2(1-p)^1=12\times 1\times p^3(1-p)^0\\\implies 3p^2-3p^3=12p^3$$
Divide throughout by $$3p^2$$, we get
$$1-p=4p\\\implies 5p=1\\\implies p=\dfrac 15$$

Mathematics

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