The Problem:
Two teams, the Exponents and the Radicals, square off in a best of 5 math hockey tournament. Once a team wins 3 games, the tournament is over.
The schedule of the tournament (for home games) goes: E-R-E-R-E
If the Exponents are playing at home, there is a 60% chance they'll win. If they are playing on the road, there is a 45% chance they'll win.
Find the probability that the Exponents win the series. Round answers to at least 4 decimal places.
3 marks for showing possible outcomes
5 marks for showing work
1 mark for a correct strategy to find final probability
1 mark for correct final probability
To find the probability that the Exponents win the series, we can consider all the possible outcomes of the tournament.
Let's label the Exponents' wins as E and the Radicals' wins as R. The series could end in any of the following outcomes:
1. E-E-E-E-E
2. E-E-E-E-R
3. E-E-E-R-E
4. E-E-E-R-R
5. E-E-R-E-E
6. E-E-R-E-R
7. E-E-R-R-E
8. E-E-R-R-R
9. E-R-E-E-E
10. E-R-E-E-R
11. E-R-E-R-E
12. E-R-E-R-R
13. E-R-R-E-E
14. E-R-R-E-R
15. E-R-R-R-E
16. E-R-R-R-R
17. R-E-E-E-E
18. R-E-E-E-R
19. R-E-E-R-E
20. R-E-E-R-R
21. R-E-R-E-E
22. R-E-R-E-R
23. R-E-R-R-E
24. R-E-R-R-R
25. R-R-E-E-E
26. R-R-E-E-R
27. R-R-E-R-E
28. R-R-E-R-R
29. R-R-R-E-E
30. R-R-R-E-R
31. R-R-R-R-E
32. R-R-R-R-R
Now, let's assign the probabilities to each outcome based on the information given. The Exponents have a 60% chance of winning at home (E) and a 45% chance of winning on the road (R).
Let's analyze outcome 1 (E-E-E-E-E) as an example:
P(E-E-E-E-E) = P(E) * P(E) * P(E) * P(E) * P(E)
= 0.6 * 0.6 * 0.6 * 0.6 * 0.6
We can calculate the probabilities for all the outcomes using the given information.
To find the final probability that the Exponents win the series, we need to sum up the probabilities of all the outcomes where the Exponents win at least three games.
So, the strategy to find the final probability is to calculate the probabilities of all the outcomes and then sum up the probabilities of the desired outcomes.
Finally, round the answer to at least 4 decimal places.
Let's calculate the probabilities for all the outcomes and find the final probability.