a) Calculate the probability distribution for getting doubles within the first seven rolls of a pair of standard dice.
b) Use these probabilities to estimate the expected number of rolls before getting doubles. How accurate is this estimate?
How do I start?
To calculate the probability distribution for getting doubles within the first seven rolls of a pair of standard dice, you need to determine the probabilities for each of the possible outcomes.
a) Here's how you can start:
Step 1: Determine the total number of possible outcomes when rolling a pair of standard dice. Since each die has six faces, there are 6 * 6 = 36 possible outcomes.
Step 2: Identify the favorable outcomes, i.e., the outcomes where you get doubles (the same number on both dice). There are six possible doubles outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).
Step 3: Calculate the probability of getting doubles in one roll. The probability of each outcome is 1/36 since the likelihood of rolling any specific outcome out of the 36 possible outcomes is 1/36.
Step 4: Calculate the probability of not getting doubles in one roll. The probability of not getting doubles is 1 - 1/36 = 35/36 because there are 35 outcomes that do not result in doubles.
Step 5: Calculate the probability of getting doubles within the first seven rolls. You can use the complement rule to find the probability of not getting doubles in any of the seven rolls and then subtract it from 1.
b) To estimate the expected number of rolls before getting doubles, you can use the concept of expected value. The expected value is calculated by multiplying each possible outcome by its corresponding probability and summing them all up.
Expected Number of Rolls = (1/36) * 1 + (35/36) * (1 + Expected Number of Rolls)
This is because there's a 1/36 probability of getting doubles on the first roll, and a 35/36 probability of not getting doubles. If you don't get doubles on the first roll, the expected number of rolls will increase by one, hence the 1 + Expected Number of Rolls.
To solve this equation, you can use algebraic techniques or iterative methods such as a recursive function or spreadsheet software.
The accuracy of this estimate depends on the assumption that the rolls are independent and that the probability of getting doubles remains constant for each roll. In reality, there might be other factors to consider, such as biases in the dice or the effect of previous rolls on subsequent rolls.
To start, let's calculate the probability distribution for getting doubles within the first seven rolls of a pair of standard dice.
The probability of rolling doubles on two fair six-sided dice is determined by the number of ways we can roll doubles divided by the total number of possible outcomes.
There are six possible outcomes for each individual die roll, so the total number of possible outcomes for two dice is 6 * 6 = 36.
The number of ways to roll doubles can be determined by looking at the possible combinations:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)
So, there are 6 possible ways to roll doubles.
Therefore, the probability of rolling doubles on any given roll is 6/36 = 1/6.
Now we can calculate the probability distribution for getting doubles within the first seven rolls.
On the first roll, the probability of rolling doubles is 1/6.
On the second roll, the probability of not rolling doubles on the first roll and then rolling doubles is (5/6) * (1/6) = 5/36.
On the third roll, the probability of not rolling doubles on the first two rolls and then rolling doubles is (5/6) * (5/6) * (1/6) = 25/216.
We can continue this process for the remaining rolls, multiplying the probabilities of not rolling doubles on the previous rolls and then rolling doubles on the current roll.
To calculate the probability distribution for the first seven rolls, we have:
- Roll 1: 1/6
- Roll 2: (5/6) * (1/6) = 5/36
- Roll 3: (5/6) * (5/6) * (1/6) = 25/216
- Roll 4: (5/6) * (5/6) * (5/6) * (1/6) = 125/1296
- Roll 5: (5/6) * (5/6) * (5/6) * (5/6) * (1/6) = 625/7776
- Roll 6: (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (1/6) = 3125/46656
- Roll 7: (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (1/6) = 15625/279936
Now let's move on to part b) where we estimate the expected number of rolls before getting doubles.