Solve the following problems by using the binomial formula.
a. If n = 4 and p = .10 , find P(x = 3) .
b. If n = 7 and p = .80 , find P(x = 4) .
c. If n = 10 and p = .60 , find P(x ¡Ý 7) .
d. If n = 12 and p = .45 , find P(5 ¡Ü x ¡Ü 7) .
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a. Well, if n = 4 and p = 0.10, P(x = 3) can be calculated using the binomial formula, which is P(x) = C(n, x) * p^x * (1-p)^(n-x).
Now let's substitute the values:
P(3) = C(4, 3) * (0.10)^3 * (1 - 0.10)^(4 - 3)
= 4 * (0.001) * (0.9)^1
= 0.0036
So, P(x = 3) is approximately 0.0036.
b. Okay, moving on. For n = 7 and p = 0.80, we can calculate P(x = 4) using the binomial formula once again.
P(4) = C(7, 4) * (0.80)^4 * (1 - 0.80)^(7 - 4)
= 35 * (0.4096) * (0.008)
= 0.114688
Hence, P(x = 4) is approximately 0.114688.
c. Alright, let's tackle the next problem. For n = 10 and p = 0.60, we need to find P(x ≥ 7).
P(x ≥ 7) = P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)
Now we can use the binomial formula to find each probability separately. I'll just give you the total result here:
P(x ≥ 7) ≈ 0.315985
So, P(x ≥ 7) is approximately 0.315985.
d. Last but not least, for n = 12 and p = 0.45, we want to find P(5 ≤ x ≤ 7).
P(5 ≤ x ≤ 7) = P(x = 5) + P(x = 6) + P(x = 7)
Using the binomial formula again, we find:
P(5 ≤ x ≤ 7) ≈ 0.215674
Therefore, P(5 ≤ x ≤ 7) is approximately 0.215674.
Hope that helps with your binomial problems! Keep calm and calculate on!
To solve these problems using the binomial formula, we need to understand the formula itself. The binomial formula is given by:
P(x) = (n C x) * p^x * (1 - p)^(n - x)
Where:
- P(x) represents the probability of getting exactly x successes in n independent trials.
- (n C x) represents the binomial coefficient, which is calculated as n! / (x! * (n - x)!), where n! is the factorial of n.
- p represents the probability of success in a single trial.
- (1 - p) represents the probability of failure in a single trial.
Now let's solve each of the given problems using this formula:
a. If n = 4 and p = 0.10, we need to find P(x = 3).
Using the binomial formula, we substitute the values into the formula:
P(3) = (4 C 3) * (0.10)^3 * (1 - 0.10)^(4 - 3)
Now, we calculate the binomial coefficient:
(4 C 3) = 4! / (3! * (4 - 3)!) = 4! / (3! * 1!) = 4
Substituting the values into the formula:
P(3) = 4 * (0.10)^3 * (0.90)^1
= 4 * 0.001 * 0.90
= 0.004 * 0.90
= 0.0036
Therefore, P(x = 3) = 0.0036.
b. If n = 7 and p = 0.80, we need to find P(x = 4).
Using the binomial formula, we substitute the values into the formula:
P(4) = (7 C 4) * (0.80)^4 * (1 - 0.80)^(7 - 4)
Calculating the binomial coefficient:
(7 C 4) = 7! / (4! * (7 - 4)!) = 35
Substituting the values into the formula:
P(4) = 35 * (0.80)^4 * (0.20)^3
= 35 * 0.4096 * 0.008
= 0.114688
Therefore, P(x = 4) = 0.114688.
c. If n = 10 and p = 0.60, we need to find P(x ≥ 7).
P(x ≥ 7) means the probability of getting 7, 8, 9, or 10 successes in 10 trials.
To find this probability, we need to calculate the individual probabilities for getting exactly 7, 8, 9, or 10 successes and then add them together.
P(x = 7) = (10 C 7) * (0.60)^7 * (1 - 0.60)^(10 - 7)
P(x = 8) = (10 C 8) * (0.60)^8 * (1 - 0.60)^(10 - 8)
P(x = 9) = (10 C 9) * (0.60)^9 * (1 - 0.60)^(10 - 9)
P(x = 10) = (10 C 10) * (0.60)^10 * (1 - 0.60)^(10 - 10)
Calculate the binomial coefficients:
(10 C 7) = 120
(10 C 8) = 45
(10 C 9) = 10
(10 C 10) = 1
Substituting the values and calculating the probabilities:
P(7) = 120 * (0.60)^7 * (0.40)^3
P(8) = 45 * (0.60)^8 * (0.40)^2
P(9) = 10 * (0.60)^9 * (0.40)^1
P(10) = 1 * (0.60)^10 * (0.40)^0
Now, add all these probabilities together:
P(x ≥ 7) = P(7) + P(8) + P(9) + P(10)
c. If n = 12 and p = 0.45, we need to find P(5 ≤ x ≤ 7).
P(5 ≤ x ≤ 7) means the probability of getting 5, 6, or 7 successes in 12 trials.
To find this probability, we need to calculate the individual probabilities for getting exactly 5, 6, and 7 successes and then add them together.
P(x = 5) = (12 C 5) * (0.45)^5 * (1 - 0.45)^(12 - 5)
P(x = 6) = (12 C 6) * (0.45)^6 * (1 - 0.45)^(12 - 6)
P(x = 7) = (12 C 7) * (0.45)^7 * (1 - 0.45)^(12 - 7)
Calculate the binomial coefficients:
(12 C 5) = 792
(12 C 6) = 924
(12 C 7) = 792
Substituting the values and calculating the probabilities:
P(5) = 792 * (0.45)^5 * (0.55)^7
P(6) = 924 * (0.45)^6 * (0.55)^6
P(7) = 792 * (0.45)^7 * (0.55)^5
Now, add all these probabilities together:
P(5 ≤ x ≤ 7) = P(5) + P(6) + P(7)
I hope this explanation helps you understand how to use the binomial formula to solve problems. Let me know if you have any further questions!
If you need to use the formula and work your problems instead of using the binomial probability table, here is the formula to use:
P(x) = nCx(p^x)[q^(n-x)]
For a:
x = 3
n = 4
p = .10
q = 1 - p = .90
I'll let you take it from here for the rest of your problems.