3. You roll a fair six-sided die 7 times. What is the probability of getting a) exactly two 4’s?
b) less than (and not equal to) two 4’s? c) two or more 4’s?
4. A carnival game has the following payouts:
payout (x) $10 $20 $30 $40
probability (p) 40% 25% 20% 15%
a) If you play the game repeatedly, what is the expected value (long-run average) of the payouts?
b) What does the carnival have to charge to make this a fair game?
To solve these probability problems, we need to use some basic concepts. Let's start with Question 3 and then move on to Question 4.
Question 3: You roll a fair six-sided die 7 times. What is the probability of:
a) Getting exactly two 4's?
b) Getting less than (and not equal to) two 4's?
c) Getting two or more 4's?
a) To find the probability of getting exactly two 4's, we need to consider the following:
- Each roll of the die has 6 equally possible outcomes.
- We need to select two of the seven rolls to be 4's, and the remaining five rolls to be anything other than 4's.
The probability of getting a 4 on any individual roll is 1/6, and since the rolls are independent (one roll does not affect the others), we need to multiply these probabilities together.
The formula for calculating combinations is used to determine the number of ways we can choose two 4's out of seven rolls. It is given by:
C(n, r) = n! / (r!(n-r)!)
In this case, we have n=7 (total number of rolls) and r=2 (number of 4's we want to get).
Using this formula, we can calculate the probability:
P(exactly two 4's) = C(7, 2) * (1/6)^2 * (5/6)^5
Calculating this expression will give us the answer.
b) To find the probability of getting less than (and not equal to) two 4's, we need to consider the total probability of not getting exactly two 4's or getting fewer than two 4's.
We can calculate this by subtracting the probability of getting exactly two 4's from 1:
P(less than two 4's) = 1 - P(exactly two 4's)
Calculating this expression will give us the answer.
c) To find the probability of getting two or more 4's, we need to consider the following:
- The probability of getting exactly two 4's (calculated in part a).
- The probability of getting three 4's, four 4's, five 4's, six 4's, or seven 4's.
To find the probabilities for each of these scenarios, we can use the same approach as in part a. Then, we add up all these probabilities to get the final answer.
Now, let's move on to Question 4.
Question 4: A carnival game has the following payouts:
- Payout (x): $10, $20, $30, $40
- Probability (p): 40%, 25%, 20%, 15%
a) To find the expected value (long-run average) of the payouts, we multiply each payout by its corresponding probability and sum the results:
Expected value = (10 * 0.4) + (20 * 0.25) + (30 * 0.2) + (40 * 0.15)
Calculating this expression will give us the answer.
b) To find what the carnival needs to charge to make this a fair game, we need to consider that a fair game implies an expected value of zero for the player.
In this case, we equate the expected value calculated in part a to zero and solve for the payout price charged by the carnival.
Expected value = (price * probability) + (price * probability) + (price * probability) + (price * probability)
By solving this equation for the price, we can find the amount the carnival needs to charge.
I hope this explanation helps you understand how to solve these probability questions. Let me know if you have any further questions!