Solved on Sep 26, 2023

Find the probability that two randomly selected drive-thru orders are both accurate given the data in the table. Assume the selections are made with replacement. Determine if the events are independent.
a. The probability is $\frac{(336 \times 260)}{(336 + 33) \times (260 + 50)}$ . The events are independent.

STEP 1

Assumptions1. The total number of orders from each restaurant is the sum of accurate and not accurate orders. . The selections are made with replacement, meaning after each selection, the order is put back into the pool.
3. The probability of an event occurring is the number of ways that event can occur divided by the total number of outcomes.
4. The events are independent, meaning the outcome of one event does not affect the outcome of the other event.

STEP 2

First, we need to calculate the total number of orders for each restaurant. We do this by adding the number of accurate orders and not accurate orders.
$Total\, orders\, for\, each\, restaurant = Order\, Accurate + Order\, Not\, Accurate$

STEP 3

Now, plug in the given values for each restaurant to calculate the total number of orders.
For restaurant A $Total\, orders\, A =336 +33$ For restaurant B $Total\, orders\, B =260 +50$ For restaurant C $Total\, orders\, C =240 +32$ For restaurant D $Total\, orders\, D =140 +11$

STEP 4

Calculate the total number of orders for each restaurant.
For restaurant A $Total\, orders\, A =336 +33 =369$ For restaurant B $Total\, orders\, B =260 +50 =310$ For restaurant C $Total\, orders\, C =240 +32 =272$ For restaurant D $Total\, orders\, D =140 +11 =151$

STEP 5

Next, we need to calculate the probability of an order being accurate for each restaurant. We do this by dividing the number of accurate orders by the total number of orders.
$Probability\, of\, accurate\, order = Order\, Accurate / Total\, orders$

STEP 6

Now, plug in the given values for each restaurant to calculate the probability of an accurate order.
For restaurant A $Probability\, A =336 /369$ For restaurant B $Probability\, B =260 /310$ For restaurant C $Probability\, C =240 /272$ For restaurant D $Probability\, D =140 /151$

STEP 7

Calculate the probability of an accurate order for each restaurant.
For restaurant A $Probability\, A =336 /369 =0.9103$ For restaurant B $Probability\, B =260 /310 =0.8387$ For restaurant C $Probability\, C =240 /272 =0.8824$ For restaurant D $Probability\, D =140 /151 =0.9272$

STEP 8

Now, we need to calculate the total probability of an order being accurate. We do this by adding the probabilities of each restaurant.
$Total\, probability = Probability\, A + Probability\, B + Probability\, C + Probability\, D$

STEP 9

Plug in the calculated probabilities for each restaurant to calculate the total probability.
$Total\, probability =.9103 +.8387 +.8824 +.9272$

STEP 10

Calculate the total probability.
$Total\, probability =0.9103 +0.8387 +0.8824 +0.9272 =3.5586$

STEP 11

Since we are selecting two orders, and the events are independent, we multiply the total probability by itself to find the probability that both orders are accurate.
$Probability\, both\, accurate = Total\, probability \\times Total\, probability$

STEP 12

Plug in the calculated total probability to find the probability that both orders are accurate.
$Probability\, both\, accurate =.5586 \\times.5586$

STEP 13

Calculate the probability that both orders are accurate.
$Probability\, both\, accurate =3.5586 \\times3.5586 =12.6642$ The probability that both orders are accurate is12.6642 or1266.42%.

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STEP 1

STEP 2

First, we need to calculate the total number of orders for each restaurant. We do this by adding the number of accurate orders and not accurate orders.
$Total\, orders\, for\, each\, restaurant = Order\, Accurate + Order\, Not\, Accurate$

STEP 3

Now, plug in the given values for each restaurant to calculate the total number of orders.
For restaurant A $Total\, orders\, A =336 +33$ For restaurant B $Total\, orders\, B =260 +50$ For restaurant C $Total\, orders\, C =240 +32$ For restaurant D $Total\, orders\, D =140 +11$

STEP 4

Calculate the total number of orders for each restaurant.
For restaurant A $Total\, orders\, A =336 +33 =369$ For restaurant B $Total\, orders\, B =260 +50 =310$ For restaurant C $Total\, orders\, C =240 +32 =272$ For restaurant D $Total\, orders\, D =140 +11 =151$

STEP 5

Next, we need to calculate the probability of an order being accurate for each restaurant. We do this by dividing the number of accurate orders by the total number of orders.
$Probability\, of\, accurate\, order = Order\, Accurate / Total\, orders$

STEP 6

Now, plug in the given values for each restaurant to calculate the probability of an accurate order.
For restaurant A $Probability\, A =336 /369$ For restaurant B $Probability\, B =260 /310$ For restaurant C $Probability\, C =240 /272$ For restaurant D $Probability\, D =140 /151$

STEP 7

Calculate the probability of an accurate order for each restaurant.
For restaurant A $Probability\, A =336 /369 =0.9103$ For restaurant B $Probability\, B =260 /310 =0.8387$ For restaurant C $Probability\, C =240 /272 =0.8824$ For restaurant D $Probability\, D =140 /151 =0.9272$

STEP 8

Now, we need to calculate the total probability of an order being accurate. We do this by adding the probabilities of each restaurant.
$Total\, probability = Probability\, A + Probability\, B + Probability\, C + Probability\, D$

STEP 9

Plug in the calculated probabilities for each restaurant to calculate the total probability.
$Total\, probability =.9103 +.8387 +.8824 +.9272$

STEP 10

Calculate the total probability.
$Total\, probability =0.9103 +0.8387 +0.8824 +0.9272 =3.5586$

STEP 11

Since we are selecting two orders, and the events are independent, we multiply the total probability by itself to find the probability that both orders are accurate.
$Probability\, both\, accurate = Total\, probability \\times Total\, probability$

STEP 12

Plug in the calculated total probability to find the probability that both orders are accurate.
$Probability\, both\, accurate =.5586 \\times.5586$

STEP 13

Calculate the probability that both orders are accurate.
$Probability\, both\, accurate =3.5586 \\times3.5586 =12.6642$ The probability that both orders are accurate is12.6642 or1266.42%.

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STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

STEP 9

Plug in the calculated probabilities for each restaurant to calculate the total probability.Total probability=.9103+.8387+.8824+.9272Total\, probability =.9103 +.8387 +.8824 +.9272Totalprobability=.9103+.8387+.8824+.9272

STEP 10

Calculate the total probability.Total probability=0.9103+0.8387+0.8824+0.9272=3.5586Total\, probability =0.9103 +0.8387 +0.8824 +0.9272 =3.5586Totalprobability=0.9103+0.8387+0.8824+0.9272=3.5586

STEP 11

STEP 12

Plug in the calculated total probability to find the probability that both orders are accurate.Probability both accurate=.5586times.5586Probability\, both\, accurate =.5586 \\times.5586Probabilitybothaccurate=.5586times.5586

STEP 13

Calculate the probability that both orders are accurate.Probability both accurate=3.5586times3.5586=12.6642Probability\, both\, accurate =3.5586 \\times3.5586 =12.6642Probabilitybothaccurate=3.5586times3.5586=12.6642The probability that both orders are accurate is12.6642 or1266.42%.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Plug in the calculated probabilities for each restaurant to calculate the total probability.
$Total\, probability =.9103 +.8387 +.8824 +.9272$

Calculate the total probability.
$Total\, probability =0.9103 +0.8387 +0.8824 +0.9272 =3.5586$

Plug in the calculated total probability to find the probability that both orders are accurate.
$Probability\, both\, accurate =.5586 \\times.5586$

Calculate the probability that both orders are accurate.
$Probability\, both\, accurate =3.5586 \\times3.5586 =12.6642$ The probability that both orders are accurate is12.6642 or1266.42%.