Question

Police report that 90% of drivers stopped on suspicion of drunk driving are given a
breath test, 11% are given a blood test, and 8% are given both.
(a) In this context, define two events A and B.
(b) Write the given information in probability notation.
(c) Explain, in this context, the meaning of P(A ∩ B).
(d) Are A and B disjoint events?
(e) Are A and B independent events?

Answer 1

Probabilities of an Events

(a).

Event A = a drunk driving are given a breath test.

Event B = a drunk driving are given a blood test.

(b).

P(A) = 90% =$\frac{90} {100} =0.9$

P(B ) = 11% = $\frac {11} {100} = 0.11$

P(Both) = $P(A\bigcap B ) = 8$% = $\frac{8} {100} = 0.08$

(c).

$P(A\bigcap B ) =$ The probability that the drivers stopped on suspicion of drunk driving are given both

breath test AND blood test.

(d).

We knows, If the Events A and B are Disjoint, we can write as

$$P(A\bigcap B) = 0$$

But, Here $P(A\bigcap B) = 0.08 \ne 0$

So, the Events A and B are not disjoint

(e).

If the events A and B are independent, then $P(A\bigcap B) = P(A) \times P(B)$

Here,

$$P(A\bigcap B ) = 0.08 \space and \space P(A) \times P(B) = 0.9 \times 0.11 = 0.099$$

So,

$$P(A\bigcap B) \ne P(A) \times P(B)$$

So, the events A and B are not Independent