what is the probability of observing three or fewer 6s when rolling a fair die twenty times?
To find the probability of observing three or fewer 6s when rolling a fair die twenty times, we need to use the concept of binomial probability. The binomial distribution is commonly used to model the number of successes in a fixed number of independent Bernoulli trials.
In this case, each roll of the fair die can be considered as a Bernoulli trial, where the success is defined as observing a 6 and the failure is defined as observing any other number. The probability of success (rolling a 6) is 1/6, since there are six equally likely outcomes (numbers 1 to 6) when rolling a fair die.
So, let's break down the problem into cases:
Case 1: Observing no 6s (denoted as X = 0)
To calculate the probability of this case, we use the binomial probability formula:
P(X = 0) = C(n, x) * p^x * q^(n-x)
where n is the number of trials, x is the number of successes, p is the probability of success, q is the probability of failure (1-p), and C(n, x) is the combination formula.
In this case, n = 20, x = 0, p = 1/6, and q = 1 - p = 5/6. Plugging these values into the formula:
P(X = 0) = C(20, 0) * (1/6)^0 * (5/6)^(20-0)
Case 2: Observing one 6 (X = 1)
Similarly, we can find the probability of observing exactly one 6 using the same formula:
P(X = 1) = C(20, 1) * (1/6)^1 * (5/6)^(20-1)
Case 3: Observing two 6s (X = 2)
Using the formula again:
P(X = 2) = C(20, 2) * (1/6)^2 * (5/6)^(20-2)
Case 4: Observing three 6s (X = 3)
Again, using the formula:
P(X = 3) = C(20, 3) * (1/6)^3 * (5/6)^(20-3)
Finally, to find the probability of observing three or fewer 6s, we sum up these probabilities:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Now, you can calculate each individual probability using a calculator or software.