there is a 1/5 (.20) probability that one type of appliance will break with in two years. find the probability that 4 out 15 such appliances willl break within that time period
Try the binomial probability formula:
P(x) = (nCx)(p^x)[q^(n-x)]
n = 15
x = 4
p = .20
q = 1 - p = .80
Substitute the values into the formula and calculate your probability.
To find the probability that exactly 4 out of 15 appliances will break within the two-year time period, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = nCk * p^k * q^(n-k)
Where:
- P(X = k) represents the probability of getting exactly k successes.
- nCk represents the number of combinations of n items taken k at a time.
- p represents the probability of success on each individual trial.
- q represents the probability of failure on each individual trial (1-p).
- k represents the number of successes we want.
In this case, we have:
n = 15 (total number of appliances)
k = 4 (number of appliances that will break)
p = 0.2 (probability that an appliance will break within two years)
q = 1 - p = 1 - 0.2 = 0.8 (probability that an appliance will not break within two years)
Now we can calculate the probability using the formula:
P(X = 4) = 15C4 * (0.2)^4 * (0.8)^(15-4)
Using the combination formula of nCr:
15C4 = 15! / (4! * (15-4)!) = (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1) = 1365
Substituting the values into the formula:
P(X = 4) = 1365 * (0.2)^4 * (0.8)^(11)
Calculating this:
P(X = 4) ≈ 0.0345
Therefore, the probability that exactly 4 out of 15 appliances will break within the two-year time period is approximately 0.0345 (or 3.45%).
To find the probability that 4 out of 15 appliances will break within the two-year time period, we can use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the combination function, also known as "n choose k," which is the number of ways to choose k successes out of n trials
- p is the probability of success in a single trial
- k is the number of successes we want to find the probability for
- n is the total number of trials
In this case, the probability of getting a break in a single appliance within two years is 1/5 (or 0.20). The number of successes we want is 4, out of a total of 15 appliances.
Let's calculate the probability using this information:
P(X = 4) = C(15, 4) * (0.20)^4 * (1 - 0.20)^(15 - 4)
Now, we need to calculate the combination function:
C(15, 4) = 15! / (4! * (15 - 4)!)
This simplifies to:
C(15, 4) = 15! / (4! * 11!)
Cross-canceling the factorials, we get:
C(15, 4) = (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1)
C(15, 4) = 1365
Substituting the values back into the main formula:
P(X = 4) = 1365 * (0.20)^4 * (1 - 0.20)^(15 - 4)
Calculating this expression will give us the probability that exactly 4 out of 15 appliances will break within the two-year time period.