Find the probability of at least 3 successes and at least 2 failures. N(trials)=10 and p=3/10
To find the probability of at least 3 successes and at least 2 failures, we can use the binomial probability formula.
The binomial probability formula is given by:
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 number of trials,
- p is the probability of success in a single trial,
- k is the number of successes in a given situation.
In this case, we want to find the probability of at least 3 successes and at least 2 failures. To do this, we need to calculate the probabilities for the following situations and add them together:
1. P(X=3): The probability of getting exactly 3 successes.
2. P(X=4): The probability of getting exactly 4 successes.
3. P(X=5): The probability of getting exactly 5 successes.
4. P(X=6): The probability of getting exactly 6 successes.
5. P(X=7): The probability of getting exactly 7 successes.
6. P(X=8): The probability of getting exactly 8 successes.
7. P(X=9): The probability of getting exactly 9 successes.
8. P(X=10): The probability of getting exactly 10 successes.
The formula for each of these situations remains the same:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
In this case,
n = 10 (number of trials)
p = 3/10 (probability of success in a single trial)
k takes values from 3 to 10.
By calculating each of these probabilities separately and summing them up, we can find the final probability.