Problem 1 (35 points). The Atlanta Falcons and the playoffs. With 5 wins and 4 l

Question

Problem 1 (35 points). The Atlanta Falcons and the playoffs. With 5 wins and 4 losses (updated it after the Sunday game with the Cowboys), meaning an up and down season, Falcons are at the moment in danger of missing the playoffs. Let's be optimistic and hope they can get into the playoffs as a wild card, which would realistically require a 10-6 record at the end of the regular season. a) (10 points) Assuming that the Falcons' record so far is quite indicative of their successes probability in the remaining games, what is the probability that they finish with a record of 10-6 which will give them a good chance to win their division b) (10 points) Since the team that is on the last wild card position, the Carolina Panthers who are at 7-3 (also updated), will not remain idle, what is the probability that the Panthers would achieve only 9 wins or less. c) (10 points) Based on your results from above, are you optimistic about the Falcons chances to win ? d) (5 points) Are there any potential concerns for the analysis conducted in parts a-c?

Explanation / Answer

(a) We assume that the performance of the Falcons so far is an indication of their future performance. Falcons have won 5 and lost 4 of their 9 games played so far.

So probability of winning = 5/9.

The number of games played for resulting in wild card result is 10+6 = 16. But 9 are already played. So games left = 16-9 = 7.

Number of wins in 7 games be X.

Then X~Binomial (n=7 , p=5/9)

To find P(X=5) = 7C5 * (5/9)5 (4/9)2 = 0.22

(b) The Carolina Panthers have a record of 7-3 i.e. 7 wins in 10 games.

So probability of winning = 7/10. Number of games left = 6.

Let Y be the no. of games won by Carolina Panthers in 6 games.

Then Y~Binomial (n=6 , p=7/10).

Now, Carolina Panthers have already won 7, total number of wins has to be <9.

So to find P(Y<=2) = 0.07.

(c) So probability that Carolina panthers will win a total of more than 9 games is 1-0.07 = 0.93 which is way greater than the prob. that Falcons end on 10-6 record. So the Falcons has a very little chance of making to the play-offs.

(d) However the above calculation is done assuming that the Falcons and the Carolina Panthers are the only teams and they act independant of one another. However there might be face to face games which will affect both their records. So ideally we should calculate the joint probability of Y<=2 and X=5. The assumption of independance o Falcons and Carolina Panthers is not the right assumption.

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