Edward C. answered • 03/25/15

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Let W (for Win) denote profit in each year and L denote loss. We are given

P(W) = 0.2 in the first year so P(L) = 0.8 in the first year

If there was a profit in the first year then we have

P(WW) = 0.2*0.7 = 0.14 = probability of profit in both years

P(WL) = 0.2*0.3 = 0.06 = probability of profit in year 1 and loss in year 2

P(LW) = 0.8*0.2 = 0.16 = probability of loss in year 1 and profit in year 2

P(LL) = 0.8*0.8 = 0.64 probability of loss in both years

Continuing to the third year we have

P(WWW) = 0.14*0.7 = 0.098

P(WWL) = 0.14*0.3 = 0.042

P(WLW) = 0.06*0.2 = 0.012

P(WLL) = 0.06*0.8 = 0.048

P(LWW) = 0.16*0.7 = 0.112

P(LWL) = 0.16*0.3 = 0.048

P(LLW) = 0.64*0.2 = 0.128

P(LLL) = 0.64*0.8 = 0.512

Of these 8 possibilities, 3 have a profit in exactly 2 of the 3 years, they are

P(WWL) + P(WLW) + P(LWW) = 0.042 + 0.012 + 0.112 = 0.166