Question

between 5:00 pm and 6:00 pm, cars arrive at mcdonald’s drive-thru at the rate of 20 cars per hour. the following formula from probability can be used to determine the probability that ( x ) cars will arrive between 5:00 pm and 6:00 pm. complete parts (a) and (b). ( p(x) = \frac{20^x e^{-20}}{x!} ), where ( x! = x cdot (x - 1) cdot (x - 2) cdot dots cdot 3 cdot 2 cdot 1 ) (a) determine the probability that ( x = 12 ) cars will arrive between 5:00 pm and 6:00 pm. ( p(12) = square ) (round to two decimal places as needed.)

Explanation:

Step1: Identify the formula and values

The formula is \( P(x)=\frac{20^{x}e^{-20}}{x!} \), and we need to find \( P(12) \), so \( x = 12 \), \( \lambda=20 \) (since the rate is 20 cars per hour).

Step2: Calculate \( 20^{12} \)

\( 20^{12}=20\times20\times\cdots\times20 \) (12 times) \( =4.096\times10^{15} \)

Step3: Calculate \( e^{-20} \)

\( e^{-20}\approx2.061153622438558\times10^{-9} \)

Step4: Calculate \( 12! \)

\( 12!=12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1 = 479001600 \)

Step5: Substitute into the formula

\( P(12)=\frac{20^{12}\times e^{-20}}{12!}=\frac{4.096\times10^{15}\times2.061153622438558\times10^{-9}}{479001600} \)

First, multiply the numerator: \( 4.096\times10^{15}\times2.061153622438558\times10^{-9}=4.096\times2.061153622438558\times10^{6}\approx8.443\times10^{6} \)

Then divide by \( 479001600 \approx4.79\times10^{8} \)

\( P(12)\approx\frac{8.443\times10^{6}}{4.79\times10^{8}}\approx0.0176 \)

Wait, maybe a better way is to use a calculator for factorial and exponential:

Using a calculator:

\( 20^{12}=4096000000000000 \)

\( e^{-20}\approx2.061153622438558\times10^{-9} \)

\( 12!=479001600 \)

So numerator: \( 4096000000000000\times2.061153622438558\times10^{-9}=4096000000000000\times2.061153622438558\div10^{9} \)

\( 4096000000000000\div10^{9}=4096000 \)

\( 4096000\times2.061153622438558\approx4096000\times2.061\approx8422896 \)

Then divide by \( 479001600 \): \( 8422896\div479001600\approx0.0176 \)

Wait, but maybe I made a mistake. Let's use the Poisson probability formula correctly.

The Poisson probability formula is \( P(x;\lambda)=\frac{e^{-\lambda}\lambda^{x}}{x!} \), where \( \lambda = 20 \), \( x = 12 \)

So \( e^{-20}\approx2.061153622438558\times10^{-9} \)

\( \lambda^{x}=20^{12}=4096000000000000 \)

\( x!=12!=479001600 \)

So \( P(12)=\frac{2.061153622438558\times10^{-9}\times4096000000000000}{479001600} \)

Calculate the numerator: \( 2.061153622438558\times10^{-9}\times4096000000000000 = 2.061153622438558\times4096000000000000\times10^{-9} \)

\( 4096000000000000\times10^{-9}=4096000 \)

\( 2.061153622438558\times4096000\approx2.061153622438558\times4.096\times10^{6}\approx8.443\times10^{6} \)

Then divide by \( 479001600\approx4.79\times10^{8} \)

\( 8.443\times10^{6}\div4.79\times10^{8}\approx0.0176 \), which is approximately 0.02 when rounded to two decimal places? Wait, no, 0.0176 is approximately 0.02? Wait, 0.0176 is closer to 0.02? Wait, 0.0176 rounded to two decimal places: the third decimal is 7, which is more than 5, so we round up the second decimal: 0.02? Wait, no, 0.0176: the first decimal is 0, second is 1, third is 7. So 0.0176 rounded to two decimal places is 0.02? Wait, no, 0.0176 is 0.01 (first decimal) and 0.0076 (second decimal part). Wait, no, decimal places: first decimal is tenths, second is hundredths. So 0.0176: the hundredths place is 1, the thousandths place is 7. So we look at the thousandths place to round the hundredths place. Since 7 >=5, we round up the hundredths place: 1 becomes 2. So 0.02.

Wait, but let's use a calculator for more accuracy.

Using a calculator:

\( e^{-20} \approx 2.061153622438558 \times 10^{-9} \)

\( 20^{12} = 4096000000000000 \)

\( 12! = 479001600 \)

So \( P(12) = \frac{4096000000000000 \times 2.061153622438558 \times 10^{-9}}{479001600} \)

Calculate numerator: \( 4096000000000000 \times 2.061153622438558 \times 10^{-9} = 4096000000000000 \times 2.061153622438558 \div 10^9 \)

\( 4096000000000000 \div 10^9 = 4096000 \)

\( 4096000 \times 2.061153622438558 = 4096000 \times 2 + 4096000 \times 0.061153622438558 \)

\( 4096000 \times 2 = 8192000 \)

\( 4096000 \times 0.061153622438558 \approx 4096000 \times 0.06 = 245760 \), \( 4096000 \times 0.001153622438558 \approx 4726 \), so total approx 245760 + 4726 = 250486. So total numerator approx 8192000 + 250486 = 8442486

Then divide by 479001600: \( 8442486 \div 479001600 \approx 0.0176 \)

So 0.0176 rounded to two decimal places is 0.02? Wait, no, 0.0176 is 0.01 (to one decimal place) and 0.0176 (to two decimal places: the second decimal is 1, third is 7, so we round up the second decimal: 1 becomes 2, so 0.02.

Wait, but maybe I made a mistake in the formula. Wait, the Poisson distribution formula is \( P(x) = \frac{\lambda^x e^{-\lambda}}{x!} \), where \( \lambda \) is the average rate. Here, the rate is 20 cars per hour, so \( \lambda = 20 \), and we want \( x = 12 \). So that's correct.

Alternatively, using a calculator for Poisson probability:

Using an online Poisson calculator, with \( \lambda = 20 \), \( x = 12 \), the probability is approximately 0.0176, which is 0.02 when rounded to two decimal places.

Answer:

\( 0.02 \)