Leaving Cert Mathematics Probability: Assume that 20% of adults in Ire

Assume that 20% of adults in Ireland jog for recreation - Leaving Cert Mathematics - Question (c) - 2022

Question (c)

Assume that 20% of adults in Ireland jog for recreation. Three adults are picked at random. Find the probability that exactly one of these adults jogs for recreation... show full transcript

Worked Solution & Example Answer:Assume that 20% of adults in Ireland jog for recreation - Leaving Cert Mathematics - Question (c) - 2022

Step 1

Find the Probability of Exactly One Adult Jogging

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Answer

To solve this, we can use the binomial probability formula, which is given by:

$P(X = k) = {n race k} p^k (1-p)^{n-k}$

Where:

$n$ is the number of trials (in this case 3 adults),
$k$ is the number of successful trials (1 adult jogging),
$p$ is the probability of success (0.2), and
$1-p$ is the probability of failure (0.8).

We need to calculate the probability for $k=1$ :

$P(X = 1) = {3 race 1} (0.2)^1 (0.8)^{3-1}$

Calculating:

Compute the binomial coefficient: ${3 race 1} = 3$
Compute the probabilities:
- $0.2^1 = 0.2$
- $0.8^2 = 0.64$

Thus, the probability is:

$P(X = 1) = 3 imes 0.2 imes 0.64 = 3 imes 0.128 = 0.384$

So, the probability that exactly one adult jogs is $P(X = 1) = 0.384$ .

Step 2

Calculate the Expected Cost of Classes for a Year

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Answer

Let $X$ be the random variable representing the number of classes Sinéad attends in a year.

The probabilities and classes per week are:

No classes: 30% chance, so $0 imes 1 = 0$
1 class a week: 60% chance, so $1 imes 52 = 52$
2 classes a week: 10% chance, so $2 imes 104 = 208$

Now, we calculate the expected value $E(X)$ :

$E(X) = 0.3 imes 0 + 0.6 imes 52 + 0.1 imes 104$

Calculating:

$E(X) = 0 + 31.2 + 10.4 = 41.6$

Now, to find the expected cost of classes at €6 per class:

$ext{Expected Cost} = E(X) imes ext{Cost per class} = 41.6 imes 6 = 249.6$

Thus, the cost of the classes for a year is approximately €250.

Assume that 20% of adults in Ireland jog for recreation - Leaving Cert Mathematics - Question (c) - 2022

Worked Solution & Example Answer:Assume that 20% of adults in Ireland jog for recreation - Leaving Cert Mathematics - Question (c) - 2022

Find the Probability of Exactly One Adult Jogging

Calculate the Expected Cost of Classes for a Year

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