In a competition Mary has a probability of $ \frac{1}{20} $ of winning, a probability of $ \frac{1}{10} $ of finishing in second place, and a probability of $ \frac{1}{4} $ of finishing in third place. If she wins the competition she gets €9000. If she comes second she gets €7000 and if she comes third she gets €3000. In all other cases she gets nothing. Each participant in the competition must pay €2000 to enter. (a) Find the expected value of Mary's loss if she enters the competition. (b) Each of the 3 prizes in the competition above is increased by the same amount (x) but the entry fee is unchanged. For example, if Mary wins the competition now, she would get $ €(9000 + x) $. Mary now expects to break even. Find the value of x.

Leaving Cert Mathematics Probability: In a competition Mary has a prob

In a competition Mary has a probability of $ \frac{1}{20} $ of winning, a probability of $ \frac{1}{10} $ of finishing in second place, and a probability of $ \frac{1}{4} $ of finishing in third place - Leaving Cert Mathematics - Question 1 - 2018

Question 1

$In-a-competition-Mary-has-a-probability-of--$-\frac{1}{20}-$-of-winning,-a-probability-of-$-\frac{1}{10}-$-of-finishing-in-second-place,-and-a-probability-of-$-\frac{1}{4}-$-of-finishing-in-third-place-Leaving Cert Mathematics-Question 1-2018.png$

In a competition Mary has a probability of $ \frac{1}{20} $ of winning, a probability of $ \frac{1}{10} $ of finishing in second place, and a probability of \( ... show full transcript

Worked Solution & Example Answer:In a competition Mary has a probability of $ \frac{1}{20} $ of winning, a probability of $ \frac{1}{10} $ of finishing in second place, and a probability of $ \frac{1}{4} $ of finishing in third place - Leaving Cert Mathematics - Question 1 - 2018

Step 1

Find the expected value of Mary's loss if she enters the competition.

96%

114 rated

Answer

To find the expected value of Mary's loss, we first need to calculate the expected value of her winnings and subtract the entry fee.

Probabilities and Prize Amounts:
- Probability of winning: ( \frac{1}{20} ) for €9000
- Probability of finishing second: ( \frac{1}{10} ) for €7000
- Probability of finishing third: ( \frac{1}{4} ) for €3000
- Probability of not placing: ( 1 - \left(\frac{1}{20} + \frac{1}{10} + \frac{1}{4} \right) = 1 - \left(\frac{1}{20} + \frac{2}{20} + \frac{5}{20} \right) = 1 - \frac{8}{20} = \frac{12}{20} )
Calculate Expected Value of Winnings: $E(X) = \left( \frac{1}{20} \times 9000 \right) + \left( \frac{1}{10} \times 7000 \right) + \left( \frac{1}{4} \times 3000 \right) + \left( \frac{12}{20} \times 0 \right)$ $E(X) = 450 + 700 + 375 + 0 = 1525$
Entry Fee: The entry fee is €2000.
Expected Loss: $\text{Expected Loss} = \text{Entry Fee} - \text{Expected Winnings} = 2000 - 1525 = 475$

Thus, the expected value of Mary's loss if she enters the competition is €475.

Step 2

Find the value of x.

99%

104 rated

Answer

In part (b), we know that Mary expects to break even. This means her expected winnings should equal her entry fee.

Updated Prizes:
- Winning prize: ( 9000 + x )
- Second prize: ( 7000 + x )
- Third prize: ( 3000 + x )
Probabilities Remain the Same:
- Probability of winning: ( \frac{1}{20} )
- Probability of second place: ( \frac{1}{10} )
- Probability of third place: ( \frac{1}{4} )
Set Up the Equation for Break-even: $E(X) = \left( \frac{1}{20} \cdot (9000 + x) \right) + \left( \frac{1}{10} \cdot (7000 + x) \right) + \left( \frac{1}{4} \cdot (3000 + x) \right)$ $2000 = \frac{9000}{20} + \frac{7000}{10} + \frac{3000}{4} + \left( \frac{x}{20} + \frac{x}{10} + \frac{x}{4} \right)$
Simplify: $2000 = 450 + 700 + 750 + \left( \frac{x}{20} + \frac{2x}{20} + \frac{5x}{20} \right)$ $2000 = 1900 + \frac{8x}{20}$ $100 = \frac{8x}{20}$
Solve for x: $x = 250$

Therefore, the value of x is €250.

In a competition Mary has a probability of \( \frac{1}{20} \) of winning, a probability of \( \frac{1}{10} \) of finishing in second place, and a probability of \( \frac{1}{4} \) of finishing in third place - Leaving Cert Mathematics - Question 1 - 2018

Worked Solution & Example Answer:In a competition Mary has a probability of \( \frac{1}{20} \) of winning, a probability of \( \frac{1}{10} \) of finishing in second place, and a probability of \( \frac{1}{4} \) of finishing in third place - Leaving Cert Mathematics - Question 1 - 2018

Find the expected value of Mary's loss if she enters the competition.

Find the value of x.

Join the Leaving Cert students using SimpleStudy...