A bag contains 25 tickets each colored either red or yellow. Red tickets are worth $.50 and yellow tickets are $5.00. If the expected value of a ticket drawn at random from this bag is $3.20, how many tickets are red?
50r + 500(25-r) = 320*25
r = 10
Well, if the expected value of a ticket is $3.20 and there are only red and yellow tickets in the bag, we can set up an equation. Let's call the number of red tickets "R" and the number of yellow tickets "Y".
The expected value can be calculated by multiplying the value of each ticket by their probabilities and adding them up. So we have:
0.5R + 5Y = 3.20
But we also know that there are a total of 25 tickets in the bag, so we can write:
R + Y = 25
Now we have a system of equations to solve. Let me put on my thinking cap...
*puts on a cap with lots of question marks on it*
Hmm, let's see. Multiplying the second equation by -0.5 and adding it to the first equation, we get:
-0.5R - 0.5Y + 0.5R + 5Y = -0.5(25) + 3.20
Simplifying that, we have:
4.5Y = -12.5 + 3.20
Now it's time to do some math!
*starts juggling numbers*
Y = (-12.5 + 3.20) / 4.5
Y = -9.3 / 4.5
Y ≈ -2.06666666667
Oh dear! It seems we've encountered a mathematical catastrophe! Negative tickets are not allowed (unless you're attending a mime convention). Let me rethink this.
*puts on a jester hat and starts juggling again*
Ah, silly me! I made a mistake. Since we can't have negative tickets, let's assume all the remaining tickets are red. That means:
R = 25 - Y
Now let's substitute this value into the first equation:
0.5(25 - Y) + 5Y = 3.20
Simplifying that, we get:
12.5 - 0.5Y + 5Y = 3.20
Combining like terms:
4.5Y = 3.20 - 12.5
4.5Y = -9.30
Dividing both sides by 4.5, we find:
Y = -9.30 / 4.5
Y ≈ -2.06666666667
Oh no! I did it again! I made the same mistake! I must be the funniest mathematician ever. Let me try this one more time.
*puts on a big red clown nose and starts juggling imaginary tickets*
Okay, let's go back to our original equation:
0.5R + 5Y = 3.20
And our second equation:
R + Y = 25
To solve this system, we can use substitution! Let's solve the second equation for R:
R = 25 - Y
Now we can substitute that into the first equation:
0.5(25 - Y) + 5Y = 3.20
Simplifying that, we get:
12.5 - 0.5Y + 5Y = 3.20
Combining like terms again:
4.5Y = 3.20 - 12.5
4.5Y = -9.30
Dividing both sides by 4.5, we find:
Y = -9.30 / 4.5
Y ≈ -2.06666666667
Oh no! I did it again! I made the same mistake twice! I must be the world's funniest mathematician. Let me take off my silly costume and get back to you with the correct answer.
To find the number of red tickets, we can set up an equation using the concept of expected value.
Let's assume that there are 'R' red tickets and 'Y' yellow tickets in the bag, where R + Y = 25.
The expected value (EV) is calculated by multiplying the value of each ticket by its probability and summing them up. In this case, we have:
EV = (value of a red ticket) * (probability of drawing a red ticket) + (value of a yellow ticket) * (probability of drawing a yellow ticket)
Given that red tickets are worth $0.50 and yellow tickets are worth $5.00, we can write the equation as:
3.20 = 0.50 * (R/25) + 5.00 * (Y/25)
Simplifying this equation gives us:
3.20 = 0.02R + 0.20Y
Now, we need to solve this equation to find the value of R.
To do this, we can solve the equations together with the constraint R + Y = 25.
Let's multiply both sides of the equation by 100 to remove the decimals:
320 = 2R + 20Y
We have the following system of equations:
2R + 20Y = 320
R + Y = 25
We can solve this system of equations using substitution, elimination, or any other method of solving simultaneous equations. Let's use the elimination method:
Multiply both sides of the second equation by 2:
2R + 2Y = 50
Now, subtract the first equation from this new equation:
(2R + 20Y) - (2R + 2Y) = 320 - 50
Simplifying, we get:
18Y = 270
Divide both sides by 18:
Y = 15
Now, substitute this value of Y back into the second equation to find R:
R + 15 = 25
R = 10
Therefore, there are 10 red tickets in the bag.
To find out how many tickets are red, we need to set up an equation based on the given information. Let's assume that there are x red tickets in the bag.
Since the expected value of a ticket drawn at random from the bag is $3.20, we can set up the equation:
(0.5 * x) + (5 * (25 - x)) = 3.20
Let's solve this equation to find the value of x:
0.5x + 125 - 5x = 3.20
-4.5x = 3.20 - 125
-4.5x = -121.8
x = (-121.8) / (-4.5)
x ≈ 27.07
Since the number of red tickets must be a whole number, we can conclude that there are 27 red tickets in the bag.