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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?

Mellie  May 15, 2015

Best Answer 

 #3
avatar+92625 
+11

 

1 2 3 4 5 6 7 8  FOUR IN A ROW
W W W L            $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$
      L W W W W
L W W W W L           $${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$
  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  L          $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$
     L W W W W W
L W W W W W  L       $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$      
  L W W W W W  L

 

 

1 2 3 4 5 6 7 8                       SIX IN A ROW
W W W W  W  L        $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$
   L  W W W W W W
L W W W W W  W  L     $${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

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

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$

 

$${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$

 

FIVE IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$

 

SIX IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

SEVEN IN A ROW  

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{7}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

EIGHT IN A ROW

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{8}}} = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

 

$${\frac{{\mathtt{32}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{16}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{729}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\frac{{\mathtt{112}}}{{\mathtt{243}}}} = {\mathtt{0.460\: \!905\: \!349\: \!794\: \!238\: \!7}}$$

Melody  May 17, 2015
 #1
avatar+86919 
+4

 

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%

 

 

CPhill  May 15, 2015
 #2
avatar+92625 
+6

I am sure it is not that simple Chris  

Melody  May 17, 2015
 #3
avatar+92625 
+11
Best Answer

 

1 2 3 4 5 6 7 8  FOUR IN A ROW
W W W L            $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$
      L W W W W
L W W W W L           $${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$
  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  L          $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$
     L W W W W W
L W W W W W  L       $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$      
  L W W W W W  L

 

 

1 2 3 4 5 6 7 8                       SIX IN A ROW
W W W W  W  L        $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$
   L  W W W W W W
L W W W W W  W  L     $${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

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

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$

 

$${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$

 

FIVE IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$

 

SIX IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

SEVEN IN A ROW  

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{7}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

EIGHT IN A ROW

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{8}}} = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

 

$${\frac{{\mathtt{32}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{16}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{729}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\frac{{\mathtt{112}}}{{\mathtt{243}}}} = {\mathtt{0.460\: \!905\: \!349\: \!794\: \!238\: \!7}}$$

Melody  May 17, 2015
 #4
avatar+1781 
+1

Thank you both, but Melody's answer was correct. Thanks guys so so much once again!!

Mellie  May 17, 2015

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