document.write( "Question 1192083: 5. The number of white corpuscles on a slide has a Poisson distribution with mean 3.6.
\n" ); document.write( "a. Find the most likely number of white corpuscles on a slide.
\n" ); document.write( "b. Calculate correct to three decima places the probability of obtaining this number.
\n" ); document.write( "c. If such two slides are prepared, what is the probability, correct to three decimal places, of obtaining at least two white corpuscles in total on the two slides?\r
\n" ); document.write( "\n" ); document.write( "6. At Foodland Supermarket 65% of customers pay by debit card. Find the probability that in a randomly selected sample of twenty customers.
\n" ); document.write( "a. Exactly five pay by debit card
\n" ); document.write( "b. More than eighteen pay by cash.
\n" ); document.write( "(give answers to 3 decimal places)
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Algebra.Com's Answer #849088 by CPhill(1987)\"\" \"About 
You can put this solution on YOUR website!
Here's the solution:\r
\n" ); document.write( "\n" ); document.write( "**Problem 1: White Corpuscles**\r
\n" ); document.write( "\n" ); document.write( "**a. Most Likely Number:**\r
\n" ); document.write( "\n" ); document.write( "For a Poisson distribution, the most likely number is the integer part of the mean (λ). In this case, λ = 3.6. Therefore, the most likely number of white corpuscles is 3.\r
\n" ); document.write( "\n" ); document.write( "**b. Probability of Obtaining the Most Likely Number:**\r
\n" ); document.write( "\n" ); document.write( "Use the Poisson probability formula:\r
\n" ); document.write( "\n" ); document.write( "P(x) = (e^(-λ) * λ^x) / x!\r
\n" ); document.write( "\n" ); document.write( "P(3) = (e^(-3.6) * 3.6^3) / 3!
\n" ); document.write( "P(3) ≈ (0.0273 * 46.656) / 6
\n" ); document.write( "P(3) ≈ 0.217\r
\n" ); document.write( "\n" ); document.write( "The probability of obtaining 3 white corpuscles is approximately 0.217.\r
\n" ); document.write( "\n" ); document.write( "**c. Probability of at Least Two White Corpuscles in Total on Two Slides:**\r
\n" ); document.write( "\n" ); document.write( "* The mean for two slides is 2 * 3.6 = 7.2.
\n" ); document.write( "* We want P(X ≥ 2). It's easier to calculate the complement: P(X ≥ 2) = 1 - P(X < 2) = 1 - [P(0) + P(1)]\r
\n" ); document.write( "\n" ); document.write( "* P(0) = (e^(-7.2) * 7.2^0) / 0! ≈ 0.0007
\n" ); document.write( "* P(1) = (e^(-7.2) * 7.2^1) / 1! ≈ 0.0051\r
\n" ); document.write( "\n" ); document.write( "* P(X ≥ 2) = 1 - (0.0007 + 0.0051)
\n" ); document.write( "* P(X ≥ 2) = 1 - 0.0058
\n" ); document.write( "* P(X ≥ 2) ≈ 0.994\r
\n" ); document.write( "\n" ); document.write( "The probability of at least two white corpuscles in total is approximately 0.994.\r
\n" ); document.write( "\n" ); document.write( "**Problem 2: Foodland Supermarket**\r
\n" ); document.write( "\n" ); document.write( "This is a binomial probability problem.\r
\n" ); document.write( "\n" ); document.write( "* n = 20 (number of customers)
\n" ); document.write( "* p = 0.65 (probability of paying by debit card)
\n" ); document.write( "* q = 1 - p = 0.35 (probability of paying by cash)\r
\n" ); document.write( "\n" ); document.write( "**a. Exactly Five Pay by Debit Card:**\r
\n" ); document.write( "\n" ); document.write( "P(X = 5) = (20C5) * p^5 * q^15
\n" ); document.write( "P(X = 5) = (20! / (5! * 15!)) * (0.65)^5 * (0.35)^15
\n" ); document.write( "P(X = 5) ≈ 15504 * 0.1160 * 1.406 x 10⁻⁸
\n" ); document.write( "P(X = 5) ≈ 0.002\r
\n" ); document.write( "\n" ); document.write( "The probability of exactly five customers paying by debit card is approximately 0.002.\r
\n" ); document.write( "\n" ); document.write( "**b. More Than Eighteen Pay by Cash:**\r
\n" ); document.write( "\n" ); document.write( "P(X > 18) = P(X = 19) + P(X = 20)\r
\n" ); document.write( "\n" ); document.write( "P(X = 19) = (20C19) * (0.65)^19 * (0.35)^1 ≈ 20 * 0.00017 * 0.35 ≈ 0.001
\n" ); document.write( "P(X = 20) = (20C20) * (0.65)^20 * (0.35)^0 ≈ 1 * 0.00037 * 1 ≈ 0.000\r
\n" ); document.write( "\n" ); document.write( "P(X > 18) ≈ 0.001 + 0.000 = 0.001\r
\n" ); document.write( "\n" ); document.write( "The probability of more than eighteen customers paying by cash is approximately 0.001.
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