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### Course: Statistics and probability > Unit 7

Lesson 7: Multiplication rule for independent events- Compound probability of independent events
- "At least one" probability with coin flipping
- Free-throw probability
- Three-pointer vs free-throw probability
- Probability without equally likely events
- Independent events example: test taking
- Die rolling probability with independent events
- Independent probability
- Probabilities involving "at least one" success
- Probability of "at least one" success

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# Probabilities involving "at least one" success

## Example 1 : Defective chips

A manufacturer of processing chips knows that $2\mathrm{\%}$ of its chips are defective in some way.

Suppose an inspector randomly selects $4$ chips for an inspection.

**Assuming the chips are independent, what is the probability that at least one of the selected chips is defective?**

Lets break this problem up into smaller pieces to understand the strategy behind solving it.

## Example 2: Surgical implants

Surgeries involving implants sometimes result in the patient's body rejecting the implant. A certain surgery has a rejection rate of $11\mathrm{\%}$ . The rest of the patients successfully accept the implant.

Assume that the results for each patient are independent.

## Example 3: Free-throws

Esther is a basketball player who makes $75\mathrm{\%}$ of the free-throws she attempts. Assume that the results of each shot are independent.

## Generalizing the strategy

In general, we can use these strategies:

or similarly,

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