++ Prep

00:00 5 marks

A discrete random variable \(X\) represents the number of defects found in a manufactured item. The probability mass function of \(X\) is partially given by:

\(P(X=1)=0.1,\quad P(X=2)=0.3,\quad P(X=3)=k,\quad P(X=4)=0.4\).

(a) Determine the value of \(k\) that makes this a valid probability mass function.

(b) Using the value of \(k\) obtained, calculate the conditional probability \(P(X=3 | X>2)\).

Part a

4 minutes 2 marks

Question

Determine the value of \(k\) that makes this a valid probability mass function.

Hint Hint

Hint

Sum all given probabilities and set their total equal to 1.

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k = 0.2

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Part b

6 minutes 3 marks

Question

Using the value of \(k\) from part (a), calculate the conditional probability \(P(X=3 | X>2)\).

Hint Hint

Hint

Recall that \(P(A|B)=\frac{P(A \cap B)}{P(B)}\).

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Complete Answer

P(X=3 | X>2) = \(0.2/0.6 = \frac{1}{3} \approx 0.333\)

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