Question

10 different names are put into the computer and one of the name is Colin. a) On Saturday, the computer chooses 2 names at random. The computer is set so the same name *can* be chosen twice. What is the probability that Colin is chosen at least once? b) On Sunday, the computer is set so that the same name *cannot* be chosen twice. Work out the probability that Colin is chosen.

Answer 1

Answer:

a. The Probability that his name will be selected AT LEAST ONCE (with replacement) = 19/100

b. The probability of selecting Colin's name (without replacement)

= 1/5

Step-by-step explanation:

If a name can be selected twice by the computer with replacement, then the probability that Colin's name will be selected at least once is - the probability that his name will be selected in the first attempt or that his name will be selected in the second attempt or that his name will be selected in both attempts.

Since Colin is just one out of ten names that are put into the computer, he has one chance out of ten to be selected and other names are seen as having a combined 9/10 chance.

a. The probability that Colin's name will be selected at the first attempt is:

P(first attempt) (with replacement)

(1/10) × 9/10

= 9/100

His name can also be selected in the second attempt.

P(second attempt)

= (9/10) × 1/10

= 9/100

He can also be selected in both attempts.

P(both attempt)

= (1/0) × 1/10

= 1/100

The probability that his name will be selected at least once = (9/100) + (9/100) + (1/100)

= 19/100

ALTERNATIVELY

The probability that Colin's name will be selected at least once can be calculated by subtracting the probability of not selecting Colin from 1.

That is, 1- P (not selecting Colin's name)

P(not selecting colin) =

9/10 × 9/10

= 81/100

1 - P(not selecting Colin's name)

1- (81/100)

= 19/100

b. The probability of selecting Colin in two attempts without replacement is:

1 - P (not selected)

= (9/10) × 8/9

= 8/10

Subtracting this probability from 1 will then give:-

1 - (8/10)

= 2/10

= 1/5

Therefore the probability of selecting colin with replacement is 19/100 and the probability of selecting him without replacement is 1/5

Answer 2

Probabilities are used to determine the chances of an event

• The probability of choosing Collin at least once is 0.19
• The probability of choosing Collin is: 0.2

The sample size is given as 10

The number of selection is given as 2

(a) The probability that Collin is chosen at least once

Collin is just one of the 10 names.

So, the probability of not choosing Collin in each selection is 9/10

In both selections, the probability of not choosing Collin is:

$\mathbf{P' = \frac{9}{10} \times \frac{9}{10}}$

$\mathbf{P' = \frac{81}{100}}$

Using the complement rule, the probability of choosing Collin at least once is:

$\mathbf{P = 1 - P'}$

So, we have:

$\mathbf{P = 1 - \frac{81}{100}}$

Take LCM

$\mathbf{P = \frac{100 -81}{100}}$

$\mathbf{P = \frac{19}{100}}$

$\mathbf{P = 0.19}$

(b) The probability that Collin is chosen

This selection indicates without replacement

The probabilities of not choosing Collin in each selection are 9/10 and 8/9, respectively

In both selections, the probability of not choosing Collin is:

$\mathbf{P' = \frac{9}{10} \times \frac{8}{9}}$

$\mathbf{P' = \frac{8}{10}}$

Using the complement rule, the probability of choosing Collin is:

$\mathbf{P = 1 - P'}$

So, we have:

$\mathbf{P = 1 - \frac{8}{10}}$

Take LCM

$\mathbf{P = \frac{10 -8}{10}}$

$\mathbf{P = \frac{2}{10}}$

$\mathbf{P = 0.2}$

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