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Step-by-step explanation

Probability of winning a game by team Alpha and betas are same

So,

Probability of winning a game by team alpha P(α)=0.6

Probability of winning a game by team betas P(β)=0.4

a)All first four game win by team alphas

 probability of this Case

=P(α)×P(α)×P(α)×P(α)

=0.6×0.6×0.6×0.6

=0.1296

b)probability that match ends after 4 game in this situation we have two  case

Case 1 = all first four game win by team alphas

Now probability of this Case

=P(α)×P(α)×P(α)×P(α)

=0.6×0.6×0.6×0.6

=0.1296

Case 2= all first four game win by team betas

Now probability of this case

=P(β)×P(β)×P(β)×P(β)

=0.4×0.4×0.4×0.4

=0.0256

So now required probability

=0.1296+0.0256

=0.1552

c)probability that the match ends after five games

In this situation we have two cases

Case 1 = 3 out of 4 game win by team alpha and than 5th game also win by team alphas

Now probability of this case

For first 4 game probability finding by binomial distribution

Formula of binomial distribution

P(X=x)=nCx × p^x × q^(n-x)

=4C3×0.6^3×0.6

=4×0.6^4

=0.5184

Now probability 5th game also won by team alphas

=0.5184×0.6

=0.31104

Case 2 = 3 out of 4 game win by team betas and than 5th game also win by team betas

For first 4 game probability

=4C3×0.4^3×0.4

=4×0.4^4

=0.1024

Now probability that 5th game also won by team betas

=0.1024×0.4

=0.04096

Now required probability

=0.31104+0.04096

=0.352