Question

use the \at least once\ rule to find the probability of the following event.

purchasing at least one winning lottery ticket out of 9 tickets when the probability of winning is 0.03 on a single ticket

the probability is .
(round to four decimal places as needed.)

Explanation:

Identify the given parameters

We are given the following values:

• Number of independent trials (tickets purchased), \(n = 9\)
• Probability of winning on a single ticket, \(p = 0.03\)

Find the probability of the complement event

Using the Probability of Events concept, we find the probability of losing on a single ticket:

$$ q = 1 - p = 1 - 0.03 = 0.97 $$

Calculate the probability of losing all trials

The probability of purchasing 0 winning tickets out of 9 is:

$$ P(\text{no wins}) = q^n = 0.97^9 $$

$$ 0.97^9 \approx 0.760228 $$

Apply the "at least once" rule

Using Theoretical Probability, the probability of winning at least once is the complement of winning zero times:

$$ P(\text{at least one win}) = 1 - P(\text{no wins}) $$

$$ P(\text{at least one win}) = 1 - 0.97^9 \approx 1 - 0.760228 = 0.239772 $$

Round to four decimal places

Rounding \(0.239772\) to four decimal places gives:

$$ P(\text{at least one win}) \approx 0.2398 $$

Answer:

Use the "at least once" rule to find the probability of the following event.

Purchasing at least one winning lottery ticket out of 9 tickets when the probability of winning is 0.03 on a single ticket

The probability is <blank>0.2398</blank>.
(Round to four decimal places as needed.)