Question

Assertion (A) : Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is \(\frac{1}{3}\)

Reason (R) : Let E and F be two events with a random experiment, then \(P(F/E) = \frac {P(E \cap F)}{P(E)}\)

Choose the correct answer out of the following choices:

(a) Both (A) and (R) are true and (R) is the correct explanation of (A)

(b) Both (A) and (R) are true, but (R) is not the correct explanation of (A)

(c) (A) is true and (R) is false

(d) (A) is false, but (R) is true

Answer 1

Correct option is : (a) Both (A) and (R) are true and (R) is the correct explanation of (A)

Consider an event of tossing two coins

Let event E : getting two heads ; E = {HH}

Let event F : getting atleast one head; F = {HH, HT, TH}

E\( \cap \)F = {H H}

Sample space (S) = {HH, HT, TH, HT}

n(S) = 4

n(E) = 1, n(F) = 3, \(\text{n} (E \cap F) = 1\)

\(P(E) = \frac{\text{n(E)}}{\text{n(S)}} = \frac{1}{4}\)

\(P(F) = \frac{\text{n(F)}}{\text{n(S)}} = \frac{3}{4}\)  …(i)

\(P(E \cap F ) = \frac{E\ \cap \ F}{n(S)}\)  …(ii)

P(getting two heads given that atleast one head comes up)

\(= \left(\frac{E}{F}\right) = \frac{P(E\ \cap \ F)}{P(F)}\)

\(= \frac{1/4}{3/4} = \frac{1}{3}\)  (using (i) and (ii)

\(\therefore \) Assertion (A) is correct and Reason (R) is the correct explanation of (A)