If you roll a 6 sided dice 6 times, what is the best prediction possible for the number of times you will roll a two?
A) 6
B) 1
C) 2
D) 0
What do you think?
Huh? Can anyone else help?
Prob(rolling a two) = 1/6
number of expected times of getting a 2 in 6 rolls
= 6(1/6) = 1
No. You are probably not going roll a 2 six times in a row.
Ms. Sue, can I have some help? I think 6 is the answer because there are 6 sides?
But only one of those sides has a two.
To find the best prediction for the number of times you will roll a two when rolling a six-sided dice six times, you need to consider the probability of rolling a two on each individual roll.
The probability of rolling a two on a single roll of a fair six-sided dice is 1/6, since there is one possible outcome (rolling a two) out of six equally likely outcomes (rolling a one, two, three, four, five, or six).
Since each roll of the dice is independent of the others, you can multiply the probabilities together to find the probability of a specific outcome occurring multiple times. In this case, you want to find the probability of rolling a two exactly x times out of six rolls.
To calculate this, you can use the binomial distribution formula, which is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes (rolling a two) out of n trials (rolling the dice).
- nCk is the binomial coefficient, which represents the number of ways to choose k items from a set of n items (in this case, the number of ways to roll a two x times out of six rolls).
- p is the probability of success (rolling a two on a single roll).
- k is the number of successes (rolling a two exactly x times).
- (1-p) is the probability of failure (rolling a number other than two on a single roll).
- (n-k) is the number of failures (rolling a number other than two exactly (6-x) times).
The best prediction possible for the number of times you will roll a two is the value of x that maximizes the probability P(X = k). You need to calculate the probabilities for each possible value of x (0, 1, 2, 3, 4, 5, and 6) and choose the one with the highest probability.
Let's go through the options:
A) 6:
The probability of rolling a two exactly six times out of six rolls is (6C6) * (1/6)^6 * (5/6)^(6-6) = (1) * (1/6)^6 * (5/6)^0 = (1/6)^6 = 1/46656, which is a very low probability.
B) 1:
The probability of rolling a two exactly one time out of six rolls is (6C1) * (1/6)^1 * (5/6)^(6-1) = (6) * (1/6) * (5/6)^5 ≈ 0.1608, which is a higher probability than in option A.
C) 2:
The probability of rolling a two exactly two times out of six rolls is (6C2) * (1/6)^2 * (5/6)^(6-2) = (15) * (1/6)^2 * (5/6)^4 ≈ 0.1613, which is slightly higher than option B.
D) 0:
The probability of rolling a two exactly zero times out of six rolls is (6C0) * (1/6)^0 * (5/6)^(6-0) = (1) * (1) * (5/6)^6 ≈ 0.3349, which is the highest probability so far.
Based on these calculations, the best prediction possible for the number of times you will roll a two is:
D) 0