Two independent events F and S are represented in the Venn diagram shown below. $P(F \cap S) = \frac{1}{4}$, $P(F \cap S') = \frac{1}{5}$, $P(S \cap F') = x$, and $P(F \cup S') = y$, where $x, y \neq 0$. Find the value of $x$ and the value of $y$. In a club there are German, Irish and Spanish children only. There are 10 Spanish children. There are twice as many Irish children as German children. They are all in a group waiting to get on a swing. One child will be selected at random to go first and will not re-join the group. Then a second child will be selected at random to go next. The probability that the first child selected will be German and that the second child selected will not be German is $rac{1}{6}$. Find how many children are in the club.

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Two independent events F and S are represented in the Venn diagram shown below - Leaving Cert Mathematics - Question 6 - 2019

Question 6

Two independent events F and S are represented in the Venn diagram shown below. $P(F \cap S) = \frac{1}{4}$, $P(F \cap S') = \frac{1}{5}$, $P(S \cap F') = x$, and $... show full transcript

Worked Solution & Example Answer:Two independent events F and S are represented in the Venn diagram shown below - Leaving Cert Mathematics - Question 6 - 2019

Step 1

Find the value of x

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Answer

We know the events F and S are independent, so:

$P(F \cap S) = P(F) \times P(S)$

From the question, we have: $P(F \cap S) = \frac{1}{4}$ $P(F \cap S') = \frac{1}{5}$

This leads to: $P(S) = 1 - P(F \cap S') = 1 - \frac{1}{5} = \frac{4}{5}$

Using the equation $P(F \cap S) = P(F) \times P(S)$ :

$P(S) = 1 - P(F \cap S) = 1 - \frac{1}{4} = \frac{3}{4}$

Set $P(F)=\frac{1}{4}$ and solve for $x$ :

$\frac{1}{4} = \frac{1}{4} \times P(S) \Rightarrow\ P(S) = \frac{5}{9} \Rightarrow x = \frac{11}{45}$

Step 2

Find the value of y

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Answer

From the earlier calculation:

$P(F \cup S') = P(F) + P(S') - P(F \cap S)$

Calculate $P(S') = 1 - P(S) = 1 - \frac{5}{9} = \frac{4}{9}$ :

Then:

$y = \frac{1}{4} + \frac{4}{9} - \frac{1}{4} = \frac{11}{45}$

Step 3

Find the total number of children in the club

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Answer

Let n be the number of German children. Then there are 2n Irish children and 10 Spanish children.

Total number of children in the club = n + 2n + 10 = 3n + 10.

The probability that the first child is German and the second is not German is given as:

$\frac{n}{3n + 10} \cdot \frac{(3n + 10 - 1)}{(3n + 10)} = \frac{1}{6}$

Setting this equation:

$\frac{n(3n + 9)}{(3n + 10)(3n + 9)} = \frac{1}{6}$

Through simplification, we find n = 5. Therefore, the total number of children:

$3(5) + 10 = 25$ .

Two independent events F and S are represented in the Venn diagram shown below - Leaving Cert Mathematics - Question 6 - 2019

Worked Solution & Example Answer:Two independent events F and S are represented in the Venn diagram shown below - Leaving Cert Mathematics - Question 6 - 2019

Find the value of x

Find the value of y

Find the total number of children in the club

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