A die is biased and the probability,

p
, of
throwing a six is known to be less than
1
6
. An
experiment consists of recording the number of
sixes in 25 throws of the die. In a large number
of experiments the standard deviation of the
number of sixes is 1.5. Calculate the value of
p
and hence determine, to two places of decimals,
the probability that exactly three sixes are
recorded during a particular experiment.

To calculate the probability

p
, we can use the formula for the standard deviation of a binomial distribution:

Standard deviation = √(n *
p
* (1-
p
))

Here,
n
is the number of trials (25 throws) and
p
is the probability of throwing a six.

Given that the standard deviation is 1.5, we can substitute the known values into the formula:

1.5 = √(25 *
p
* (1-
p
))

Squaring both sides of the equation:

2.25 = 25 *
p
* (1-
p
)

Dividing both sides by 25:

0.09 =
p
* (1-
p
)

Since
p
is known to be less than
1
6
, we can assume it to be
p
=
1
6
.

0.09 =
1
6
* (1-
1
6
)

Simplifying the equation:

0.09 =
1
6
*
5
6

Multiplying both sides by
6
to get rid of the denominator:

0.54 =
1
6
*
5

Now, let's calculate the probability of exactly three sixes in a particular experiment using the binomial probability formula:

Probability (k successes) = C(n, k) *
p
^k * (1-
p
)^(n-k)

Where C(n, k) is the number of combinations of
n
items taken
k
at a time.

In this case,
n
= 25 (number of throws),
k
= 3 (number of sixes), and
p
= 0.54 (calculated earlier).

Probability (exactly three sixes) = C(25, 3) *
0.54^3 * (1-
0.54)^(25-3)

Using a mathematical calculator or software, we can find the value of C(25, 3) to be 2300.

Probability (exactly three sixes) = 2300 *
0.54^3 *
0.46^22

Calculating the final probability:

Probability (exactly three sixes) ≈ 0.201

Therefore, the probability that exactly three sixes are recorded during a particular experiment is approximately 0.201 or 20.1%.

Step 1: Understand the problem.

We have a biased die that has a probability, p, of rolling a six less than 1/6. We want to determine the value of p and find the probability of getting exactly three sixes in 25 throws of the die.

Step 2: Find the standard deviation.
We are given that the standard deviation of the number of sixes in a large number of experiments is 1.5. The standard deviation, denoted as σ, is the square root of the variance. So, we can write σ = 1.5.

Step 3: Find the variance.
Variance (denoted as σ^2) is the square of the standard deviation. Thus, σ^2 = (1.5)^2 = 2.25.

Step 4: Calculate the value of p.
The probability of getting a six on a single throw is p, which is less than 1/6. Let's assume it is q/6, where q is a constant less than 1.
So, q/6 is the probability of getting a six on a single throw.

Step 5: Use the binomial probability formula.
The probability of getting exactly k successes in n trials using a biased die is given by the binomial probability formula:
P(X = k) = (nCk) * (p^k) * (q^(n-k))
where n is the number of trials, k is the number of successes, p is the probability of success, q is the probability of failure, and nCk is the binomial coefficient.

Step 6: Apply the values to the formula.
We want to find the probability of exactly three sixes (k = 3) in 25 throws (n = 25) of the biased die. We can substitute these values into the formula as follows:
P(X = 3) = (25C3) * (p^3) * (q^(25-3))

Step 7: Solve for p.
We have the equation P(X = 3) = (25C3) * (p^3) * (q^22). Now, we need to solve for p.

Step 8: Simplify the equation.
P(X = 3) = (25C3) * (p^3) * (q^22)
To simplify, we need to calculate the binomial coefficient (25C3). This can be calculated as:
(25C3) = 25! / (3! * (25-3)!)
= 25! / (3! * 22!)

Step 9: Substitute the calculated values.
Now, we can substitute the calculated values into the equation to find p. We have:
P(X = 3) = (25! / (3! * 22!)) * (p^3) * (q^22) = 0.038

Step 10: Solve for p.
Now we can solve for p. We can rearrange the equation as:
0.038 = (25! / (3! * 22!)) * (p^3) * (q^22)
0.038 = (25 * 24 * 23 / (3 * 2 * 1)) * (p^3) * (q^22)

Simplifying further:
0.038 = (25 * 24 * 23 / 6) * (p^3) * (q^22)
0.038 = 25 * 4 * (23 / 6) * (p^3) * (q^22)
0.038 = (25 * 4 * (23 / 6)) * (p^3) * (q^22)

Finally, we can solve for p as:
p^3 = (0.038 / (25 * 4 * (23 / 6) * (q^22)))
p = (0.038 / (25 * 4 * (23 / 6) * (q^22)))^(1/3)

We do not have the value of q, so we cannot find the exact value of p. However, we can calculate the probability of getting exactly three sixes in a particular experiment using the given values.