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)![]() ![]() ![]() You can put this solution on YOUR website! \n" ); document.write( "Let's compute the probability of getting exactly 3 hits\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We will use this value later. Let A = 0.12853471870976\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "-----------------------------------------------------\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We will use this value later. Let B = 0.02702150336512\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "-----------------------------------------------------\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Repeat for k = 5. Keep n and p the same.\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We will use this value later. Let C = 0.002272262782976\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "-----------------------------------------------------\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We can now find the probability of getting at least 3 hits\r \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "-----------------------------------------------------\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The final answer is approximately 0.15782848485786 \n" ); document.write( "Round however you need to.\r \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |