Leaving Cert Mathematics Probability: In a large population 1 in 8 of

In a large population 1 in 8 of the people play tennis - Leaving Cert Mathematics - Question (b) - 2020

Question (b)

In a large population 1 in 8 of the people play tennis. (i) Four people are chosen at random from the population. What is the probability that the fourth person cho... show full transcript

Worked Solution & Example Answer:In a large population 1 in 8 of the people play tennis - Leaving Cert Mathematics - Question (b) - 2020

Step 1

Four people are chosen at random from the population. What is the probability that the fourth person chosen is the only one to play tennis?

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Answer

To find the probability that the fourth person is the only one who plays tennis, we need to consider the following:

The probability that the first three people do not play tennis is:

$P(No Tennis) = (1 - P(Tennis))^3 = \left( \frac{7}{8} \right)^3$
The probability that the fourth person does play tennis is:

$P(Tennis) = \frac{1}{8}$
Therefore, the combined probability can be calculated as follows:

$P(Fourth = Tennis) = \left( \frac{7}{8} \right)^3 \times \frac{1}{8}$

Calculating this gives:

$= \frac{343}{4096} \approx 0.08374$

Step 2

Three people are chosen at random from the population. What is the probability that exactly two of them play tennis?

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Answer

To determine the probability that exactly two out of three people play tennis, we can use the binomial probability formula:

The number of ways to choose 2 people from 3 is:

$\binom{3}{2} = 3$
The probability that 2 people play tennis and 1 does not is:

$P(Exactly 2 Tennis) = \binom{3}{2} \times (P(Tennis))^2 \times (P(No Tennis))^1$

So we have:

Probability of tennis = ( P(Tennis) = \frac{1}{8} )
Probability of not playing tennis = ( P(No Tennis) = \frac{7}{8} )

Therefore, substituting these values gives:

$P(Exactly 2 Tennis) = 3 \times \left( \frac{1}{8} \right)^2 \times \left( \frac{7}{8} \right)$

Calculating this yields:

$= \frac{21}{512} \approx 0.04101$

In a large population 1 in 8 of the people play tennis - Leaving Cert Mathematics - Question (b) - 2020

Worked Solution & Example Answer:In a large population 1 in 8 of the people play tennis - Leaving Cert Mathematics - Question (b) - 2020

Four people are chosen at random from the population. What is the probability that the fourth person chosen is the only one to play tennis?

Three people are chosen at random from the population. What is the probability that exactly two of them play tennis?

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