From the
Exploring Data website - http://curriculum.qed.qld.gov.au/kla/eda/
© Education Queensland, 1997
Solution - the probability that the teacher is the first to win 5 games
To determine the pattern I will calculate the probability that the teacher wins in exactly 7 games, and then apply this pattern to the other possibilities.
To win in exactly 7 games, the teacher has to win 4 of the first 6 games, and then win game 7.
P(teacher wins 4 games from 6) = 6C4 (2/3)4 (1/3)2.
P(teacher wins 4 games from 6, and game 7) = [ 6C4 (2/3)4 (1/3)2 ] * (2/3) = 6C4 (2/3)5 (1/3)2.
The Calculations
P(teacher wins in exactly 5 games) = (2/3)5 = .1317
P(teacher wins in exactly 6 games) = 5C4 (2/3)5 (1/3) = .2195
P(teacher wins in exactly 7 games) = 6C4 (2/3)5 (1/3)2 = .2195
P(teacher wins in exactly 8 games) = 7C4 (2/3)5 (1/3)3 = .1707
P(teacher wins in exactly 9 games) = 8C4 (2/3)5 (1/3)4 = .1138
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P(teacher wins) = 0.8551
P(student wins) = 1 - P(teacher wins) = = 0.1449
Excel Spreadsheet Calculations
If the teacher wants to increase the probability of becoming champion, then the number of games to be won will have to increase. If the champion is the first to win 10 games then the probability that the teacher becomes the champion rises to 0.935. This output from an Excel spreadsheet shows all of the individual calculations in on the next page, for both 5 wins and 10 wins.
Probability that teacher (student) is the first to win 5 games |
|||||
Teacher |
Student |
||||
No. games |
5 |
No. games |
5 |
||
P(wins game) |
0.667 |
P(wins game) |
0.333 |
||
Probability teacher wins in: |
Probability student wins in: |
||||
5 |
0.132 |
5 |
0.004 |
||
6 |
0.219 |
6 |
0.014 |
||
7 |
0.219 |
7 |
0.027 |
||
8 |
0.171 |
8 |
0.043 |
||
9 |
0.114 |
9 |
0.057 |
||
Total |
0.855 |
Total |
0.145 |
||
Probability that teacher (student) is the first to win 10 games. |
|||||
Teacher |
Student |
||||
No. games |
10 |
No. games |
10 |
||
P(wins game) |
0.667 |
P(wins game) |
0.333 |
||
Probability teacher wins in: |
Probability student wins in: |
||||
10 |
0.017 |
10 |
0.0000 |
||
11 |
0.058 |
11 |
0.0001 |
||
12 |
0.106 |
12 |
0.0004 |
||
13 |
0.141 |
13 |
0.0011 |
||
14 |
0.153 |
14 |
0.0024 |
||
15 |
0.143 |
15 |
0.0045 |
||
16 |
0.119 |
16 |
0.0074 |
||
17 |
0.091 |
17 |
0.0113 |
||
18 |
0.064 |
18 |
0.0161 |
||
19 |
0.043 |
19 |
0.0214 |
||
Total |
0.935 |
Total |
0.0648 |
||