Question

The chances that a visit to a primary health centre (PHC) results in neither lab
work, nor referred to a specialist is 35%. Out of those coming to a PHC, 30% are
referred to a specialist, and 40% require lab work. Find the probability that a visit
to a PHC results in both lab work and referral to a specialist.

Answer 1

Answer on Question #55136 – Math – Statistics and Probability

The chances that a visit to a primary health centre (PHC) results in neither lab work, nor referred to a specialist is 35%. Out of those coming to a PHC, 30% are referred to a specialist, and 40% require lab work. Find the probability that a visit to a PHC results in both lab work and referral to a specialist.

Solution

Consider the following events: $R =$ "referral to a specialist", $L =$ "lab work".

We shall use the two formulae:

$$P[R \cup L] = P[R] + P[L] - P[R \cap L] \quad \text{(rule of addition)}$$ $$1 - P[\overline{(R \cup L)}] = P[R \cup L] \quad \text{(rule of subtraction)}$$

We want to find

$$\begin{aligned} P[R \cap L] &= P[R] + P[L] - P[R \cup L] = P[R] + P[L] - 1 + P[\overline{(R \cup L)}] = \\ &= P[R] + P[L] - 1 + P[\bar{R} \cap \bar{L}] = 0.30 + 0.40 - 1 + 0.35 = 0.05. \end{aligned}$$

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