Three computers are chosen at random from an inventory of Dell and Acer computers for a bookstore display. Assume the same number each brand of computers is in stock. Find the prbability that
(a) All three will be Acers.
(b) Exactly two will be Dells.
(c) At most one will be Acer.
Write your answers in exact, simplified form.
Part: 0 / 3
Part 1 of 3
(a) The probability that all three will be Acers is
Answer
Final Answer: The probability that all three computers will be Acers is \(\boxed{\frac{1}{8}}\).
Step 1 ：We are choosing 3 computers from an inventory of Dell and Acer computers. We are assuming that the number of each brand of computers is the same. The probability that all three will be Acers is the product of the probability that the first computer is an Acer, the probability that the second computer is an Acer given that the first one was an Acer, and the probability that the third computer is an Acer given that the first two were Acers.
Step 2 ：Let's denote the total number of computers as \(n\) and the number of Acer computers as \(a\). Since the number of Dell and Acer computers is the same, \(a = n/2\).
Step 3 ：The probability that the first computer is an Acer is \(a/n = (n/2)/n = 1/2\).
Step 4 ：After the first Acer computer is chosen, there are \(n-1\) computers left, and \(a-1\) of them are Acers. So the probability that the second computer is an Acer given that the first one was an Acer is \((a-1)/(n-1) = ((n/2)-1)/(n-1)\).
Step 5 ：After the first two Acer computers are chosen, there are \(n-2\) computers left, and \(a-2\) of them are Acers. So the probability that the third computer is an Acer given that the first two were Acers is \((a-2)/(n-2) = ((n/2)-2)/(n-2)\).
Step 6 ：The probability that all three will be Acers is the product of these three probabilities.
Step 7 ：The simplified expression for the probability that all three computers will be Acers is \((0.125*n - 0.5)/(n - 1)\). However, since we are assuming that the number of each brand of computers is the same, \(n\) is actually a large number. As \(n\) approaches infinity, the \(-0.5\) in the numerator and the \(-1\) in the denominator become negligible. Therefore, the probability that all three computers will be Acers approaches \(0.125\) or \(1/8\) as \(n\) approaches infinity.
Step 8 ：Final Answer: The probability that all three computers will be Acers is \(\boxed{\frac{1}{8}}\).
