Determine the probability of 3 sixes in 5 tosses on a fair die.

Prob(a 6) = 1/6 , prob(not a 6) = 5/6

so in 5 tosses you want that 3 times
Prob(of your event)
= C(5,3) (1/6)^3 (5/6)^2
= 10(1/216)(25/36)
= 125/3888 or appr .03215

Well, here's a little math magic for you. The probability of rolling a six on a fair die is 1/6. So in any given toss, the chance of rolling anything other than a six is 5/6.

To find the probability of getting 3 sixes in 5 tosses, we'll need to use a combination formula. There are 5 tosses, so we need to choose 3 of them to be sixes. The formula is:

P = (nCr) * (p^r) * (q^(n-r))

where P is the probability, nCr is the combination formula, p is the probability of success, q is the probability of failure, n is the number of trials, and r is the number of successes we're looking for.

Plugging in the numbers, we get:

P = (5C3) * (1/6)^3 * (5/6)^2

Calculating this gives us approximately 0.03237, or about 3.24%.

So, the probability of getting 3 sixes in 5 tosses on a fair die is fairly low, but hey, stranger things have happened! Maybe you'll get lucky and roll some sixes!

To determine the probability of getting 3 sixes in 5 tosses on a fair die, we need to calculate the probability of getting exactly 3 sixes in any order.

The probability of getting a six on a single toss of a fair die is 1/6. The probability of not getting a six on a single toss is 5/6.

To calculate the probability of getting exactly 3 sixes, we can use the binomial probability formula:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

where:
- P(X=k) is the probability of getting exactly k successes,
- n is the total number of trials,
- k is the number of successes,
- p is the probability of success on a single trial, and
- (nCk) is the number of combinations of n things taken k at a time.

In our case, n = 5 (number of tosses), k = 3 (number of sixes), and p = 1/6 (probability of getting a six).

Let's calculate it step-by-step:

Step 1: Calculate the number of combinations (nCk).
(nCk) = 5C3 = (5!)/(3!(5-3)!) = (5*4*3!)/(3!*2*1) = 10.

Step 2: Calculate the probability of getting exactly 3 sixes.
P(X=3) = 10 * (1/6)^3 * (5/6)^(5-3)
= 10 * (1/6)^3 * (5/6)^2
= 10 * (1/216) * (25/36)
= 250/7776
β‰ˆ 0.0321543

Therefore, the probability of getting exactly 3 sixes in 5 tosses on a fair die is approximately 0.0321543.

To determine the probability of getting exactly 3 sixes in 5 tosses of a fair die, we need to calculate the number of successful outcomes (getting 3 sixes) and divide it by the total number of possible outcomes.

Step 1: Calculate the number of successful outcomes.
The number of successful outcomes is given by the binomial coefficient formula, which represents the number of ways to choose k items from a set of n items. In this case, we need to choose 3 tosses to be sixes (successes) from the 5 tosses, so we can calculate it as:

nCr = n! / (r! * (n-r)!),

where "!" denotes a factorial operation.

Here, n = 5 (total number of tosses) and r = 3 (number of desired sixes).

nCr = 5! / (3! * (5-3)!),
= 5! / (3! * 2!),
= (5 * 4 * 3!) / (3! * 2),
= (5 * 4) / 2,
= 10.

So, there are 10 different ways to obtain exactly 3 sixes in 5 tosses.

Step 2: Calculate the total number of possible outcomes.
In each toss, the die can show any of the six numbers (1, 2, 3, 4, 5, or 6). Since we have 5 tosses, the total number of outcomes is calculated as:

total outcomes = 6^5,
= 7776.

Step 3: Calculate the probability.
To find the probability, we need to divide the number of successful outcomes (10) by the total number of possible outcomes (7776):

probability = 10 / 7776,
= 0.0013 (rounded to four decimal places).

Therefore, the probability of getting exactly 3 sixes in 5 tosses of a fair die is approximately 0.0013 (or 0.13%).