raffle with a grand prize $500., 2 second place prizes of $100., 4 third place prizes of $50.. the raffle offers 250 tickets for $4. each. what is probability model , the expected value of a single ticket and the target of $200.to be raised can they expect to raise this much ?
To determine the probability model, expected value, and whether the target of $200 can be raised, let's break down the given information step by step:
1. Probability Model:
In this raffle, we have a total of 250 tickets. To find the probability of winning each type of prize, we need to calculate the probability of winning that prize divided by the total number of tickets. Here's how to do it:
- Probability of winning the grand prize: 1/250 (there's only one grand prize)
- Probability of winning a second-place prize: 2/250 (there are two second-place prizes)
- Probability of winning a third-place prize: 4/250 (there are four third-place prizes)
- Probability of not winning any prize: (250 - 1 - 2 - 4)/250 = 243/250
The probability model for this raffle is:
- Grand Prize: 1/250
- Second Place: 2/250
- Third Place: 4/250
- No Prize: 243/250
2. Expected Value of a Single Ticket:
The expected value of a single ticket is calculated by multiplying each prize amount by its corresponding probability and summing them up. Here's how to calculate it:
- Expected value of the grand prize: $500 * (1/250) = $2
- Expected value of a second-place prize: $100 * (2/250) = $0.80
- Expected value of a third-place prize: $50 * (4/250) = $0.80
- Expected value of no prize: $0 * (243/250) = $0
Now, sum up these expected values:
$2 + $0.80 + $0.80 + $0 = $3.60
Therefore, the expected value of a single ticket is $3.60.
3. Raising $200:
To determine whether they can raise $200, we need to consider the total revenue generated from selling the tickets. Since each ticket costs $4 and they are selling 250 tickets, the total revenue can be calculated as follows:
Total revenue = Number of tickets * Price per ticket
Total revenue = 250 * $4 = $1,000
As the raffle generates $1,000 in revenue, exceeding the target of $200, they can indeed expect to raise at least that amount.
In summary:
- Probability model:
- Grand Prize: 1/250
- Second Place: 2/250
- Third Place: 4/250
- No Prize: 243/250
- Expected value of a single ticket: $3.60
- They can expect to raise at least $200 as the total revenue generated from selling the tickets is $1,000.
cost = 500 + 200 + 200 = 900
revenue = 250 * 4 = 1000
obviously you can not raise 200
there are 1 + 2 + 4 = 7 winning tickets out of the 250
chance of winning 500 = 1/250
chance of winning 100 = 1/125
chance of winning 50 = 1/62.5
chance of winning anything at all = 7/250
total won = 900
total number of tickets = 250
so mean (expected) value of ticket = 900/250 = $3.60