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Problem 10

# In Exercises $$1-10,$$ you are performing five independent Bernoulli trials with $$p=.1$$ and $$q=.9 .$$ Calculate the probability of the stated outcome. Check your answer using technology. [HINT: See Quick Example $$8 .$$ At most three successes

Expert verified
The probability of at most three successes in five independent Bernoulli trials with $$p=0.1$$ and $$q=0.9$$ is approximately 0.99954.
See the step by step solution

## Step 1: Probabilities for each outcome

Suppose the number of successes is denoted by k. We need to calculate the probabilities of each outcome, i.e., $$k=0,1,2,3$$. For each k, we use the binomial probability formula: $P(k)=\binom{n}{k} p^k q^{n-k}$ where n is the number of trials, p is the probability of success, and q is the probability of failure.

## Step 2: 0 Successes

For 0 successes, we plug in $$k=0$$. $P(0)=\binom{5}{0} \times 0.1^{0} \times 0.9^{5-0}=1 \times 1\times 0.9^5=0.59049$

## Step 3: 1 Success

For 1 success, we plug in $$k=1$$. $P(1)=\binom{5}{1} \times 0.1^{1} \times 0.9^{5-1}=5 \times 0.1\times 0.9^4=0.32805$

## Step 4: 2 Successes

For 2 successes, we plug in $$k=2$$. $P(2)=\binom{5}{2} \times 0.1^{2} \times 0.9^{5-2}=10 \times 0.1^2\times 0.9^3=0.0729$

## Step 5: 3 Successes

For 3 successes, we plug in $$k=3$$. $P(3)=\binom{5}{3} \times 0.1^{3} \times 0.9^{5-3}=10 \times 0.1^3\times 0.9^2=0.0081$

## Step 6: Summing the Probabilities

Now we sum up the probabilities for each outcome. $P(\text{At most 3 successes})=P(0)+P(1)+P(2)+P(3)=0.59049+0.32805+0.0729+0.0081=0.99954$ The probability of at most three successes is approximately 0.99954.

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