Problem 1 (35 points). The Atlanta Braves and the playoffs With 51 wins and 40 l

Question

Problem 1 (35 points). The Atlanta Braves and the playoffs With 51 wins and 40 losses, meaning a better season than the last, the Braves are at the moment in control of their destiny' to get into the playoffs. Ideally, they would like to maintain win the division and avoid getting into playoffs as a wild card (they now have the second slot as a wild card). a) (10 points) Assuming that the Braves record so far is quite indicative of their successes probability in the remaining games, what is the probability that they win 5 of their next 8 games. b) (10 points) Since the team that the team Braves is chasing (Philadelphia Phillies) will not remain idle (now at 52 wins and 40 losses), what is the probability that they would win 3 games or less in their next 7 games (so that the Braves would take the lead if the scenario from part a hold up. c (10 points) Based on your results from above, are you optimistic about the Bravert chances to take their lead? d) (5 points) Are there any potential concerns for the analysis conducted in parts a-c?

Explanation / Answer

(a)

Winning probability for Braves so far, p = Total wins/Total matches = 51/(51+40) = 0.56

Since there is no other data given, we assume that the winning probability for this team for the remaining matches is equal to 0.56

So, modeling this is a Binomial distribution with parameters:

n = 8, p = 0.56

Let X denote the number of games won. We need to calculate P(X=5)

Using Binomial pdf formula:

P(X=x) = nCx*(p^x)*((1-p)^(n-x))

So,

P(X=5) = 8C5*(0.56^5)*((1-0.56)^(8-5)) = 0.262

(b)

Winning probability for Philadelphia so far, p = Total wins/Total matches = 52/(52+40) = 0.565

Since there is no other data given, we assume that the winning probability for this team for the remaining matches is equal to 0.565

So, modeling this is a Binomial distribution with parameters:

n = 7, p = 0.565

Let X denote the number of games won. We need to calculate P(X<=3)

P(X<=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

Using Binomial pdf formula:

P(X=x) = nCx*(p^x)*((1-p)^(n-x))

So,

P(X=0) = 7C0*(0.565^0)*((1-0.565)^(7-0)) = 0.003

P(X=1) = 7C1*(0.565^1)*((1-0.565)^(7-1)) = 0.027

P(X=2) = 7C2*(0.565^2)*((1-0.565)^(7-2)) = 0.104

P(X=3) = 7C3*(0.565^3)*((1-0.565)^(7-3)) = 0.226

So,

P(X<=3) = 0.003+0.027+0.104+0.226 = 0.36

(c)

In order for the Braves team to take lead, both the events must happen.

So,

P(Braves takes the lead) = P(Braves wins 5 out of their next 8 matches)*P(Philadelphia wins at most 3 matches out of their next 7 matches) = 0.262*0.36 = 0.094

So, there is a 9.4% chance that Braves will take the lead.

(d)

In calculating above probabilities, we have assumed that probability of winning remains constant for the teams in each of their remaining games. This might not hold true in practical cases because after taking a lead the teams tend to lose focus as well. So this condition might not hold in actual practice.

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