Question

use the data in the following table, which lists drive - thru order accuracy at popular fast food chains. assume that orders are randomly selected from those included in the table.
if two orders are selected, find the probability that they are both accurate. complete parts (a) and (b) below.
a. assume that the selections are made with replacement. are the events independent?
the probability is 0.7607. the events are independent.
(do not round until the final answer. round to four decimal places as needed)
b. assume that the selections are made without replacement. are the events independent?
the probability is (square). the events (square) independent.
(do not round until the final answer. round to four decimal places as needed)

Explanation:

Step1: Calculate the total number of orders

First, find the total number of orders.

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Step2: Calculate the number of accurate orders

Find the number of accurate orders: \(323+276 + 247+123=969\)

Step3: Calculate the probability without replacement

The probability that the first order is accurate is \(P_1=\frac{969}{1111}\).
Since the first order is not replaced, for the second - order, the number of accurate orders is \(969 - 1\) and the total number of orders is \(1111-1\).
The probability that the second order is accurate given the first is accurate is \(P_2=\frac{969 - 1}{1111-1}=\frac{968}{1110}\)
The probability that both orders are accurate without replacement is \(P = P_1\times P_2=\frac{969}{1111}\times\frac{968}{1110}\)

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Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\).
Let \(A\) be the event that the first order is accurate (\(P(A)=\frac{969}{1111}\)) and \(B\) be the event that the second order is accurate.
\(P(A)\times P(B)=\frac{969}{1111}\times\frac{969}{1111}=\frac{969^{2}}{1111^{2}}=\frac{938961}{1234321}\approx0.7607
eq0.7606\) (when not - replaced). So the events are not independent.

Answer:

The probability is \(0.7606\). The events are not independent.