document.write( "Question 1181224: A shoe store´s records show that 30% of the customers purchase by credit cards. This is 20 customers purchased shoes from the store.
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document.write( "a) Find the probability that at most 3 of the customers used a credit card.
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document.write( "b) What is the probability that at least 3 customers but not more than 6 used a credit card?
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document.write( "c) What is the expected number of customers to use credit cards?
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Algebra.Com's Answer #850069 by CPhill(1959)![]() ![]() You can put this solution on YOUR website! Here's how to solve this binomial probability problem:\r \n" ); document.write( "\n" ); document.write( "**a) Probability that at most 3 customers used a credit card:**\r \n" ); document.write( "\n" ); document.write( "This means we want the probability of 0, 1, 2, or 3 customers using a credit card. We'll use the binomial probability formula:\r \n" ); document.write( "\n" ); document.write( "P(x) = (nCx) * p^x * (1-p)^(n-x)\r \n" ); document.write( "\n" ); document.write( "Where:\r \n" ); document.write( "\n" ); document.write( "* n = number of trials (customers) = 20 \n" ); document.write( "* x = number of successes (customers using credit card) \n" ); document.write( "* p = probability of success (customer using credit card) = 0.3 \n" ); document.write( "* nCx = \"n choose x\" (the binomial coefficient)\r \n" ); document.write( "\n" ); document.write( "We need to calculate P(0), P(1), P(2), and P(3) and then add them together.\r \n" ); document.write( "\n" ); document.write( "* P(0) = (20C0) * (0.3)^0 * (0.7)^20 ≈ 0.0008 \n" ); document.write( "* P(1) = (20C1) * (0.3)^1 * (0.7)^19 ≈ 0.0068 \n" ); document.write( "* P(2) = (20C2) * (0.3)^2 * (0.7)^18 ≈ 0.0278 \n" ); document.write( "* P(3) = (20C3) * (0.3)^3 * (0.7)^17 ≈ 0.0716\r \n" ); document.write( "\n" ); document.write( "P(at most 3) = P(0) + P(1) + P(2) + P(3) ≈ 0.0008 + 0.0068 + 0.0278 + 0.0716 ≈ 0.107\r \n" ); document.write( "\n" ); document.write( "**b) Probability that at least 3 customers but not more than 6 used a credit card:**\r \n" ); document.write( "\n" ); document.write( "This means we want the probability of 3, 4, 5, or 6 customers using a credit card. We already have P(3). Now we calculate P(4), P(5), and P(6):\r \n" ); document.write( "\n" ); document.write( "* P(4) = (20C4) * (0.3)^4 * (0.7)^16 ≈ 0.1304 \n" ); document.write( "* P(5) = (20C5) * (0.3)^5 * (0.7)^15 ≈ 0.1789 \n" ); document.write( "* P(6) = (20C6) * (0.3)^6 * (0.7)^14 ≈ 0.1916\r \n" ); document.write( "\n" ); document.write( "P(at least 3 but not more than 6) = P(3) + P(4) + P(5) + P(6) ≈ 0.0716 + 0.1304 + 0.1789 + 0.1916 ≈ 0.5725\r \n" ); document.write( "\n" ); document.write( "**c) Expected number of customers to use credit cards:**\r \n" ); document.write( "\n" ); document.write( "The expected value of a binomial distribution is given by:\r \n" ); document.write( "\n" ); document.write( "E(x) = n * p \n" ); document.write( "E(x) = 20 * 0.3 \n" ); document.write( "E(x) = 6\r \n" ); document.write( "\n" ); document.write( "So, we expect 6 customers to use credit cards. \n" ); document.write( " \n" ); document.write( " |