A scratch card cost €5 and has two possible prizes. There is a 10% chance you get €5 back and a 1% chance you get €100. in all other cases, you get nothing. what are the expectation and standard deviation?
To find the expectation and standard deviation of the scratch card, we need to calculate the expected value and variance first.
1. Calculate the expected value:
The expected value (or mean) is a measure of the average outcome. It is calculated by multiplying each possible outcome by its probability and summing them all up.
Expected value = (Prize 1 * Probability 1) + (Prize 2 * Probability 2) + ... + (Prize n * Probability n)
In this case, we have two possible prizes:
Prize 1: €5 with a probability of 10% (0.10)
Prize 2: €100 with a probability of 1% (0.01)
Expected value = (€5 * 0.10) + (€100 * 0.01) = €0.50 + €1.00 = €1.50
So, the expected value of the scratch card is €1.50.
2. Calculate the variance:
Variance measures the spread of possible outcomes from the expected value. To calculate variance, we need to find the squared difference between each outcome and the expected value, multiply it by its probability, and sum them up.
Variance = (Outcome 1 - Expected value)^2 * Probability 1 + (Outcome 2 - Expected value)^2 * Probability 2 + ... + (Outcome n - Expected value)^2 * Probability n
In this case, the outcomes are: €5 (Prize 1), €100 (Prize 2), and €0 (No prize).
Variance = (€5 - €1.50)^2 * 0.10 + (€100 - €1.50)^2 * 0.01 + (€0 - €1.50)^2 * 0.89
= (€3.50)^2 * 0.10 + (€98.50)^2 * 0.01 + (-€1.50)^2 * 0.89
= €12.25 * 0.10 + €9,702.25 * 0.01 + €2.25 * 0.89
= €1.225 + €97.0225 + €2.0025
= €100.25
So, the variance of the scratch card is €100.25.
3. Calculate the standard deviation:
Standard deviation is the square root of the variance. It measures the variability or dispersion of the possible outcomes.
Standard deviation = sqrt(Variance)
Standard deviation = sqrt(€100.25) ≈ €10.01
Therefore, the expectation of the scratch card is €1.50 and the standard deviation is approximately €10.01.