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Question
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
Solution
The probability distribution of X is
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0 | k | 4k | 2k | k |
The given distribution is a probability distribution.
\[\therefore \sum_{} p_i = 1\]
⇒ 0 + k + 4k + 2k + k = 1
⇒8k = 1
⇒ k = 0.125
(i) P(getting admission in exactly one college) = P(X = 1) = k = 0.125
(ii) P(getting admission in at most 2 colleges) = P( X ≤ 2) = 0 + k + 4k = 5k = 0.625
(iii) P(getting admission in atleast 2 colleges) = P( X ≥ 2) = 4k + 2k + k = 7k = 0.875
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