Probability problem - A and B are college teams in overtime

You will need to solve 2 equations in 2 unknowns.In summary, the probabilities for A winning, B winning, and a tie after the first round are 0.345, 0.436, and 0.219 respectively. To find the probability that A wins, you can use recursion or solve a system of equations. To find the expected number of rounds, you can use a recursive equation and take expectations.
  • #1
bslnerd00
1
0

Homework Statement



A and B are college football teams that have gone into overtime.

In the first round A will go first with the following possible outcomes: no score; 3 points; 6 points; 7 points; 8 points; a turnover where B wins (note in this case the game ends immediately). The probabilities of these happening are: .2, .3, .1, .3, .09, .01.

B then follows with the following conditional outcomes:
if A scored 0–B ties with probability .1; B wins with probability .88; A wins with probability .02.
if A scored 3–B ties with probability .3; B wins with probability .6; A wins with probability .1.
if A scored 6–B ties with probability .01; B wins with probability .4; A wins with probability .59.
if A scored 7–B ties with probability .3; B wins with probability .1; A wins with probability .6.
if A scored 8–B ties with probability .2; A wins with probability .8
If the teams are tied after the first round, they go to a second round and continue until a team wins.

a) Find the probability that: A wins in the first round; B wins in the first round; they’re tied after the first round.
b) Find the probability that A wins.
c) Find the expected number of rounds.

Homework Equations


none

The Attempt at a Solution


a) P(A wins) = 0.345, P(B wins) = 0.436, P(tie) = 0.219

Hi, I found out the probabilities each time wins. I can't figure out the answers to B or C however. Any ideas?

Thanks
 
Physics news on Phys.org
  • #2
A wins if A wins on the first round OR if the first round ends in a tie and A wins on the second round OR if the first two rounds both end in a tie and A wins on the third round OR ...

See if you can write that mathematically. Hint: What is ##\sum_{n=0}^{\infty} x^n## ? (This has a nice simple expression).The expected number of rounds is 1*P(game ends on round 1) + 2*P(ends on round 2) + 3*P(ends on round 3) + ... .

See if you can write that mathematically. Hint: What is ##\sum_{n=0}^{\infty} n x^n## ? (This too has a nice simple expression).
 
  • #3
bslnerd00 said:

Homework Statement



A and B are college football teams that have gone into overtime.

In the first round A will go first with the following possible outcomes: no score; 3 points; 6 points; 7 points; 8 points; a turnover where B wins (note in this case the game ends immediately). The probabilities of these happening are: .2, .3, .1, .3, .09, .01.

B then follows with the following conditional outcomes:
if A scored 0–B ties with probability .1; B wins with probability .88; A wins with probability .02.
if A scored 3–B ties with probability .3; B wins with probability .6; A wins with probability .1.
if A scored 6–B ties with probability .01; B wins with probability .4; A wins with probability .59.
if A scored 7–B ties with probability .3; B wins with probability .1; A wins with probability .6.
if A scored 8–B ties with probability .2; A wins with probability .8
If the teams are tied after the first round, they go to a second round and continue until a team wins.

a) Find the probability that: A wins in the first round; B wins in the first round; they’re tied after the first round.
b) Find the probability that A wins.
c) Find the expected number of rounds.

Homework Equations


none

The Attempt at a Solution


a) P(A wins) = 0.345, P(B wins) = 0.436, P(tie) = 0.219

Hi, I found out the probabilities each time wins. I can't figure out the answers to B or C however. Any ideas?

Thanks

Besides the method suggested by DH, you can look at it recursively. Let a, b and t be the one-round probabilities that A wins, B wins, or there is a tie.

Just before starting, let x = probability that A wins eventually. If A or B win in round 1 we are done; otherwise (if there is a tie) we start again. So we get x = a + t*x; can you see why?

Similarly, let N = number of rounds until a win. If someone wins on round 1 we have N = 1; otherwise we start again and have N = 1 + N', where N' has the same probability distribution as N; that is, it is an independent, probabilistic copy of N. So: N = 1 if no tie; N = 1 + N' if tie. Take expectations to get an equation ford EN (noting that EN' = EN).
 

1. What is the probability of team A winning in overtime?

The probability of team A winning in overtime depends on various factors such as the team's performance, individual player skills, and game strategy. Without knowing these specific details, it is difficult to accurately determine the probability.

2. Is there a way to calculate the probability of both teams winning in overtime?

Yes, the probability of both teams winning in overtime can be calculated using a mathematical formula called the binomial distribution. However, this requires knowing the probability of each team winning in regular time and the number of overtimes played.

3. How does the probability change if team A has a higher win percentage than team B?

If team A has a higher win percentage than team B, it means that team A is more likely to win in regular time. This can also translate to a higher probability of winning in overtime. However, the exact change in probability would depend on the margin of win percentage between the two teams.

4. Can we predict the outcome of the game based on the probability of winning in overtime?

No, the probability of winning in overtime is just one factor among many that can influence the outcome of a game. Other factors such as team dynamics, player injuries, and luck can also play a significant role in determining the final result.

5. How does the probability of winning in overtime affect betting odds for the game?

The probability of winning in overtime can affect betting odds, but it is not the only factor taken into consideration. Bookmakers also consider other factors such as team rankings, game location, and recent performance when determining betting odds for a game.

Similar threads

  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
717
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
954
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
Replies
4
Views
626
  • Precalculus Mathematics Homework Help
Replies
7
Views
791
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
31
Views
3K
  • Calculus and Beyond Homework Help
Replies
9
Views
5K
Back
Top