document.write( "Question 1161121: A high school baseball player has a 0.296 batting average. In one game, he gets 5 at bats. What is the probability he will get at least 3 hits in the game? \n" ); document.write( "
Algebra.Com's Answer #784595 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "Let's compute the probability of getting exactly 3 hits\r
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\n" ); document.write( "\n" ); document.write( "p = 0.296 is the probability of success = probability of getting a hit
\n" ); document.write( "k = number of hits we want = 3
\n" ); document.write( "n = sample size = number of at bats (aka batting attempts) = 5\r
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\n" ); document.write( "\n" ); document.write( "Compute the binomial coefficient
\n" ); document.write( "Use the combination formula
\n" ); document.write( "n C k = (n!)/(k!*(n-k)!)
\n" ); document.write( "5 C 3 = (5!)/(3!*(5-3)!)
\n" ); document.write( "5 C 3 = (5!)/(3!*2!)
\n" ); document.write( "5 C 3 = (5*4*3!)/(3!*2!)
\n" ); document.write( "5 C 3 = (5*4)/(2!)
\n" ); document.write( "5 C 3 = (5*4)/(2*1)
\n" ); document.write( "5 C 3 = 20/2
\n" ); document.write( "5 C 3 = 10\r
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\n" ); document.write( "\n" ); document.write( "This is then useful to compute the binomial probability
\n" ); document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
\n" ); document.write( "P(X = 3) = (5 C 3)*(0.296)^(3)*(1-0.296)^(5-3)
\n" ); document.write( "P(X = 3) = (5 C 3)*(0.296)^(3)*(0.704)^(2)
\n" ); document.write( "P(X = 3) = (10)*(0.296)^(3)*(0.704)^2
\n" ); document.write( "P(X = 3) = (10)*(0.025934336)*(0.495616)
\n" ); document.write( "P(X = 3) = 0.12853471870976
\n" ); document.write( "This is the approximate probability of getting exactly 3 hits\r
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\n" ); document.write( "\n" ); document.write( "We will use this value later. Let A = 0.12853471870976\r
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\n" ); document.write( "\n" ); document.write( "Keeping n = 5 and p = 0.296 the same, we will compute the probability when k = 4. This represents the probability of getting exactly 4 hits\r
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\n" ); document.write( "\n" ); document.write( "n C k = (n!)/(k!*(n-k)!)
\n" ); document.write( "5 C 4 = (5!)/(4!*(5-4)!)
\n" ); document.write( "5 C 4 = (5!)/(4!*1!)
\n" ); document.write( "5 C 4 = (5*4!)/(4!*1!)
\n" ); document.write( "5 C 4 = (5)/(1!)
\n" ); document.write( "5 C 4 = (5)/(1)
\n" ); document.write( "5 C 4 = 5/1
\n" ); document.write( "5 C 4 = 5\r
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\n" ); document.write( "\n" ); document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
\n" ); document.write( "P(X = 4) = (5 C 4)*(0.296)^(4)*(1-0.296)^(5-4)
\n" ); document.write( "P(X = 4) = (5 C 4)*(0.296)^(4)*(0.704)^(1)
\n" ); document.write( "P(X = 4) = (5)*(0.296)^(4)*(0.704)^1
\n" ); document.write( "P(X = 4) = (5)*(0.007676563456)*(0.704)
\n" ); document.write( "P(X = 4) = 0.02702150336512\r
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\n" ); document.write( "\n" ); document.write( "We will use this value later. Let B = 0.02702150336512\r
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\n" ); document.write( "\n" ); document.write( "Repeat for k = 5. Keep n and p the same.\r
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\n" ); document.write( "\n" ); document.write( "n C k = (n!)/(k!*(n-k)!)
\n" ); document.write( "5 C 5 = (5!)/(5!*(5-5)!)
\n" ); document.write( "5 C 5 = (5!)/(5!*0!)
\n" ); document.write( "5 C 5 = (1)/(1*0!)
\n" ); document.write( "5 C 5 = (1)/(1*1)
\n" ); document.write( "5 C 5 = (1)/(1)
\n" ); document.write( "5 C 5 = 1\r
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\n" ); document.write( "\n" ); document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
\n" ); document.write( "P(X = 5) = (5 C 5)*(0.296)^(5)*(1-0.296)^(5-5)
\n" ); document.write( "P(X = 5) = (5 C 5)*(0.296)^(5)*(0.704)^(0)
\n" ); document.write( "P(X = 5) = (1)*(0.296)^(5)*(0.704)^0
\n" ); document.write( "P(X = 5) = (1)*(0.002272262782976)*(1)
\n" ); document.write( "P(X = 5) = 0.002272262782976\r
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\n" ); document.write( "\n" ); document.write( "We will use this value later. Let C = 0.002272262782976\r
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\n" ); document.write( "\n" ); document.write( "We can now find the probability of getting at least 3 hits\r
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\n" ); document.write( "\n" ); document.write( "Add up the values of A,B and C we found earlier
\n" ); document.write( "P(at least 3 hits) = P(exactly 3 hits)+P(exactly 4 hits)+P(exactly 5 hits)
\n" ); document.write( "P(at least 3 hits) = A + B + C
\n" ); document.write( "P(at least 3 hits) = 0.12853471870976 + 0.02702150336512 + 0.002272262782976
\n" ); document.write( "P(at least 3 hits) = 0.15782848485786\r
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\n" ); document.write( "\n" ); document.write( "The final answer is approximately 0.15782848485786
\n" ); document.write( "Round however you need to.\r
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