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

Discrete Mathematics

(a).

Let the Event A = Drivers given the breath test for drunk driving

Event B = Drivers given the blood test for drunk driving

(b)

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

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

P(A ∩ B)= 8% = 0.08

(c).

P(A ∩ B)= The probability that the drivers stopped on suspicion of drunk driving are both given both breath test

and blood test

(d)

If P(A ∩ B)= 0, then A and B are disjoint events .

Here, P(A ∩ B)= 0.08

So, A and B are not disjoint events.

(e)

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

P(A ∩ B)= 0.08

$P(A) \times P(B) = 0.9 \times 0.11 = 0.099$

$P(A ∩ B) \ne P(A) \times P(B)$

Answer: So, A and B are not independent events.