My basketball team has eight games next month. We're pretty good; we have a 2/3 chance to win in each game. What is the probability we have a winning streak of at least four games at some point during the month?
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | FOUR IN A ROW |
W | W | W | W | L | 2×(23)4×(13)=32243=0.1316872427983539 | |||
L | W | W | W | W | ||||
L | W | W | W | W | L | 3×(23)4×(13)2=16243=0.065843621399177 | ||
L | W | W | W | W | L | |||
L | W | W | W | W | L |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | FIVE IN A ROW |
W | W | W | W | W | L | 2×(23)5×(13)=64729=0.0877914951989026 | ||
L | W | W | W | W | W | |||
L | W | W | W | W | W | L | 2×(23)5×(13)2=642187=0.0292638317329675 | |
L | W | W | W | W | W | L |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | SIX IN A ROW |
W | W | W | W | W | W | L | 2×(23)6×(13)=1282187=0.0585276634659351 | |
L | W | W | W | W | W | W | ||
L | W | W | W | W | W | W | L | (23)6×(13)2=646561=0.0097546105776558 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | SEVEN IN A ROW |
W | W | W | W | W | W | W | L | |
L | W | W | W | W | W | W | W |
I TRIED TO PUT ALL THIS INTO A TABLE BUT THE TABLE DID NOT COPE VERY WELL
FOUR IN A ROW
2×(23)4×(13)=32243=0.1316872427983539
3×(23)4×(13)2=16243=0.065843621399177
FIVE IN A ROW
2×(23)5×(13)=64729=0.0877914951989026
2×(23)5×(13)2=642187=0.0292638317329675
SIX IN A ROW
2×(23)6×(13)=1282187=0.0585276634659351
(23)6×(13)2=646561=0.0097546105776558
SEVEN IN A ROW
2×(23)7×(13)=2566561=0.0390184423106234
EIGHT IN A ROW
(23)8=2566561=0.0390184423106234
32243+16243+64729+642187+1282187+646561+2566561+2566561=112243=0.4609053497942387
Probabilty of a winning streak of 4 = (2/3)^4 = 16/81
Probabilty of a winning streak of 5 = (2/3)^5 = 32/243
Probabilty of a winning streak of 6 = (2/3)^6 = 64/729
Probabilty of a winning streak of 7 = (2/3)^7 = 128/2187
Probabilty of a winning streak of 8 = (2/3)^8 = 256/6561
So the total probability of winning four or more in a row is
16 / 81 + 32 / 243 + 64 / 729 + 128 / 2187 + 256 / 6561 = 3376 / 6561 = about 51.46%
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | FOUR IN A ROW |
W | W | W | W | L | 2×(23)4×(13)=32243=0.1316872427983539 | |||
L | W | W | W | W | ||||
L | W | W | W | W | L | 3×(23)4×(13)2=16243=0.065843621399177 | ||
L | W | W | W | W | L | |||
L | W | W | W | W | L |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | FIVE IN A ROW |
W | W | W | W | W | L | 2×(23)5×(13)=64729=0.0877914951989026 | ||
L | W | W | W | W | W | |||
L | W | W | W | W | W | L | 2×(23)5×(13)2=642187=0.0292638317329675 | |
L | W | W | W | W | W | L |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | SIX IN A ROW |
W | W | W | W | W | W | L | 2×(23)6×(13)=1282187=0.0585276634659351 | |
L | W | W | W | W | W | W | ||
L | W | W | W | W | W | W | L | (23)6×(13)2=646561=0.0097546105776558 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | SEVEN IN A ROW |
W | W | W | W | W | W | W | L | |
L | W | W | W | W | W | W | W |
I TRIED TO PUT ALL THIS INTO A TABLE BUT THE TABLE DID NOT COPE VERY WELL
FOUR IN A ROW
2×(23)4×(13)=32243=0.1316872427983539
3×(23)4×(13)2=16243=0.065843621399177
FIVE IN A ROW
2×(23)5×(13)=64729=0.0877914951989026
2×(23)5×(13)2=642187=0.0292638317329675
SIX IN A ROW
2×(23)6×(13)=1282187=0.0585276634659351
(23)6×(13)2=646561=0.0097546105776558
SEVEN IN A ROW
2×(23)7×(13)=2566561=0.0390184423106234
EIGHT IN A ROW
(23)8=2566561=0.0390184423106234
32243+16243+64729+642187+1282187+646561+2566561+2566561=112243=0.4609053497942387