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Answer the following about the probability distribution.
Let denote the number of hours you study during a randomly selected school day. The probability…
Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values '$x$' has the following form, where '$k$' is some unknown constant.
$P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\ kx, & \text{if } x = 1 \text{ or } 2 \\ k(5-x), & \text{if } x = 3 \text{ or } 4 \\ 0, & \text{otherwise} \end{cases}$
(a)[1.0]
Find the value of '$k$'.
(b)[3.0]
What is the probability that you study:
(i) at least two hours?
(ii) exactly two hours?
(iii) at most 2 hours?
Answer
Answer (a)
AIThe probabilities add up to $1$: $0.1 + k + 2k + 2k + k = 1$.
$0.1 + 6k = 1$
Hence $k = 0.15$.
Final answer: $0.15$
Answer (b)
AI(i) $P(X \ge 2) = 2k + 2k + k = 5k = 0.75$.
(ii) $P(X = 2) = 2k = 0.30$.
(iii) $P(X \le 2) = 0.1 + k + 2k = 0.1 + 0.45 = 0.55$.
Hence the probabilities are $0.75$, $0.30$ and $0.55$.
Final answer: (i) $0.75$; (ii) $0.30$; (iii) $0.55$
From ISC 2018 Specimen Mathematics Paper 1, question 10.