Use the binomial expression (p+q)n
to calculate a binomial distribution with n = 5 and p = 0.3.
(0.3 + 0.7)^5 = 0.973 = 97.3%
To calculate a binomial distribution using the binomial expression (p+q)^n, you need to substitute the values of n and p into the formula.
In this case, n = 5 and p = 0.3.
The formula becomes (0.3 + q)^5.
To calculate the terms in the binomial expansion, you can use the binomial theorem or Pascal's triangle. The binomial theorem states that for any positive integer power of (p+q), the terms can be found using the formula:
C(n, r) * p^r * q^(n-r)
Where C(n, r) is the binomial coefficient, given by:
C(n, r) = n! / (r!(n-r)!)
Let's calculate the binomial distribution with n = 5 and p = 0.3:
When r = 0:
C(5, 0) * 0.3^0 * q^(5-0) = 1 * 1 * q^5 = q^5
When r = 1:
C(5, 1) * 0.3^1 * q^(5-1) = 5 * 0.3 * q^4 = 1.5 * q^4
When r = 2:
C(5, 2) * 0.3^2 * q^(5-2) = 10 * 0.3^2 * q^3 = 0.45 * q^3
When r = 3:
C(5, 3) * 0.3^3 * q^(5-3) = 10 * 0.3^3 * q^2 = 0.27 * q^2
When r = 4:
C(5, 4) * 0.3^4 * q^(5-4) = 5 * 0.3^4 * q = 0.18 * q
When r = 5:
C(5, 5) * 0.3^5 * q^(5-5) = 1 * 0.3^5 * q^0 = 0.00243
Therefore, the binomial distribution with n = 5 and p = 0.3 is:
q^5 + 1.5 * q^4 + 0.45 * q^3 + 0.27 * q^2 + 0.18 * q + 0.00243.
To calculate a binomial distribution with n = 5 and p = 0.3, you can use the binomial expression (p+q)^n.
The general formula to calculate the probability for each value of k is:
P(k) = C(n, k) * p^k * q^(n-k)
where:
- P(k) is the probability of getting k successes
- C(n, k) is the binomial coefficient, also known as n choose k, which can be calculated as C(n, k) = n! / (k! * (n-k)!)
- p is the probability of success
- q is the probability of failure (q = 1 - p)
- n is the total number of trials
Now let's calculate the binomial distribution for n = 5 and p = 0.3:
Step 1: Calculate q
q = 1 - p = 1 - 0.3 = 0.7
Step 2: Calculate the probabilities for each value of k (0 to 5)
P(0) = C(5, 0) * 0.3^0 * 0.7^5
P(1) = C(5, 1) * 0.3^1 * 0.7^4
P(2) = C(5, 2) * 0.3^2 * 0.7^3
P(3) = C(5, 3) * 0.3^3 * 0.7^2
P(4) = C(5, 4) * 0.3^4 * 0.7^1
P(5) = C(5, 5) * 0.3^5 * 0.7^0
Step 3: Calculate the binomial coefficients
C(5, 0) = 5! / (0! * (5-0)!) = 1
C(5, 1) = 5! / (1! * (5-1)!) = 5
C(5, 2) = 5! / (2! * (5-2)!) = 10
C(5, 3) = 5! / (3! * (5-3)!) = 10
C(5, 4) = 5! / (4! * (5-4)!) = 5
C(5, 5) = 5! / (5! * (5-5)!) = 1
Step 4: Substitute the values into the formula to calculate the probabilities:
P(0) = 1 * 0.3^0 * 0.7^5 = 0.16807
P(1) = 5 * 0.3^1 * 0.7^4 = 0.36015
P(2) = 10 * 0.3^2 * 0.7^3 = 0.3087
P(3) = 10 * 0.3^3 * 0.7^2 = 0.1323
P(4) = 5 * 0.3^4 * 0.7^1 = 0.02835
P(5) = 1 * 0.3^5 * 0.7^0 = 0.00243
So, the binomial distribution for n = 5 and p = 0.3 is:
P(0) = 0.16807
P(1) = 0.36015
P(2) = 0.3087
P(3) = 0.1323
P(4) = 0.02835
P(5) = 0.00243