Question

Problem #9: Let E and F be events whose probabilities are given in each case below. In which cases are E and F independent? (i) Pr(E) = 0.9, Pr(F) = 0.8 and Pr(FUE) = 0.99. (ii) Pr(E) = 0.4, Pr(F) = 0.5 and Pr(FUE) = 0.69. (iii) Pr(E) = 0.3, Pr(F) = 0.1 and Pr(FUE) = 0.37.

Answer 1

Two events A and B are said to be independent if P(A) * P(B) = P(A $\bigcap$ B)

P(A $\bigcap$ B) is calculated from the below equation

P(A U B) = P(A) + P(B) - P(A $\bigcap$ B)

Question (i)

P(E) = 0.9, P(F) = 0.8, P(E U F) = 0.99

P(E U F) = P(E) + P(F) - P(E $\bigcap$ F)

0.99 = 0.9 + 0.8 - P(E $\bigcap$ F)

P(E $\bigcap$ F) = 1.7 - 0.99

= 0.71

P(E) * P(F) = 0.9 * 0.8

= 0.72

P(E) * P(F) $\neq$ P(E $\bigcap$ F)

Hence E and F are not independent

Question (ii)

P(E) = 0.4, P(F) = 0.5, P(E U F) = 0.69

P(E U F) = P(E) + P(F) - P(E $\bigcap$ F)

0.69 = 0.4 + 0.5 - P(E $\bigcap$ F)

P(E $\bigcap$ F) = 0.9 - 0.69

= 0.21

P(E) * P(F) = 0.4 * 0.5

= 0.2

P(E) * P(F) $\neq$ P(E $\bigcap$ F)

Hence E and F are not independent

Question (iii)

P(E) = 0.3, P(F) = 0.1, P(E U F) = 0.37

P(E U F) = P(E) + P(F) - P(E $\bigcap$ F)

0.37 = 0.3 + 0.1 - P(E $\bigcap$ F)

P(E $\bigcap$ F) = 0.4 - 0.37

= 0.03

P(E) * P(F) = 0.3 * 0.1

= 0.03

P(E) * P(F) $=$ P(E $\bigcap$ F)

Hence E and F are independent

So only in Case (iii) E and F are independent