Let's solve the questions one by one.
Question 1:
Probability that Nikki rolls a sum of 4 with two dice, then rolls a sum of 7 with two dice.
Step 1: Find the probability of rolling a sum of 4.
When rolling two six-sided dice, the possible combinations that yield a sum of 4 are:
- (1, 3)
- (2, 2)
- (3, 1)
There are 3 combinations that result in a sum of 4.
The total number of outcomes when rolling two dice is \(6 \times 6 = 36\).
So, the probability \(P(A)\) of rolling a sum of 4 is: \[ P(A) = \frac{\text{Number of combinations for sum of 4}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12} \]
Step 2: Find the probability of rolling a sum of 7.
The combinations that yield a sum of 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
There are 6 combinations that result in a sum of 7.
So, the probability \(P(B)\) of rolling a sum of 7 is: \[ P(B) = \frac{\text{Number of combinations for sum of 7}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]
Step 3: Calculate the combined probability.
Since the two events (rolling a sum of 4 and then rolling a sum of 7) are independent, \[ P(A \text{ and } B) = P(A) \times P(B) = \left(\frac{1}{12}\right) \times \left(\frac{1}{6}\right) = \frac{1}{72} \]
Question 2:
Probability that Nikki flips heads and the dice sum to 11.
Step 1: Find the probability of flipping heads.
The probability \(P(H)\) of flipping heads on a fair coin is: \[ P(H) = \frac{1}{2} \]
Step 2: Find the probability of rolling a sum of 11.
The combinations that yield a sum of 11 are:
- (5, 6)
- (6, 5)
There are 2 combinations that result in a sum of 11.
So, the probability \(P(C)\) of rolling a sum of 11 is: \[ P(C) = \frac{\text{Number of combinations for sum of 11}}{\text{Total outcomes}} = \frac{2}{36} = \frac{1}{18} \]
Step 3: Calculate the combined probability.
Since the two events (flipping heads and getting a sum of 11) are independent, \[ P(H \text{ and } C) = P(H) \times P(C) = \left(\frac{1}{2}\right) \times \left(\frac{1}{18}\right) = \frac{1}{36} \]
Question 3:
Probability of landing on a gray section of the spinner and rolling a sum of 9 with two dice.
Step 1: Find the probability of landing on a gray section of the spinner.
Assuming there are \(n\) sections on the spinner and \(g\) of those sections are gray: \[ P(G) = \frac{g}{n} \]
(You need to provide the specific values for \(g\) and \(n\) to calculate this probability).
Step 2: Find the probability of rolling a sum of 9.
The combinations that yield a sum of 9 are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)
There are 4 combinations that result in a sum of 9.
So, the probability \(P(D)\) of rolling a sum of 9 is: \[ P(D) = \frac{\text{Number of combinations for sum of 9}}{\text{Total outcomes}} = \frac{4}{36} = \frac{1}{9} \]
Step 3: Calculate the combined probability.
Since the two events (landing on gray and rolling a sum of 9) are independent, \[ P(G \text{ and } D) = P(G) \times P(D) = \left(\frac{g}{n}\right) \times \left(\frac{1}{9}\right) \]
Replace \(g\) and \(n\) with the actual values to get the final answer.
Summary of Answers:
- The probability of rolling a sum of 4 then a sum of 7 is \(\frac{1}{72}\).
- The probability of flipping heads and rolling a sum of 11 is \(\frac{1}{36}\).
- The probability of landing on a gray section and rolling a sum of 9 is \(\left(\frac{g}{n}\right) \times \left(\frac{1}{9}\right)\), where \(g\) and \(n\) need to be specified.