To convert the given odds for each event to the probability that the event will occur, we use the formula for converting odds to probability:
[tex]\[ P = \frac{1}{1 + \text{Odds}} \][/tex]
We will apply this formula step-by-step for each event.
1. A random lottery ticket will win the jackpot:
- Odds = 195,130,342
- Using the formula:
[tex]\[ P_{lottery} = \frac{1}{1 + 195,130,342} \][/tex]
[tex]\[ P_{lottery} = \frac{1}{195,130,343} \][/tex]
- The probability, therefore, is approximately:
[tex]\[ P_{lottery} = 5.124779594119813 \times 10^{-9} \][/tex]
2. An adult will be struck by lightning during a year:
- Odds = 824,473
- Using the formula:
[tex]\[ P_{lightning} = \frac{1}{1 + 824,473} \][/tex]
[tex]\[ P_{lightning} = \frac{1}{824,474} \][/tex]
- The probability, therefore, is approximately:
[tex]\[ P_{lightning} = 1.2128945242663807 \times 10^{-6} \][/tex]
3. An adult will file for personal bankruptcy during a year:
- Odds = 155.6
- Using the formula:
[tex]\[ P_{bankruptcy} = \frac{1}{1 + 155.6} \][/tex]
[tex]\[ P_{bankruptcy} = \frac{1}{156.6} \][/tex]
- The probability, therefore, is approximately:
[tex]\[ P_{bankruptcy} = 0.006385696040868455 \][/tex]
4. A person collects stamps:
- Odds = 58.04
- Using the formula:
[tex]\[ P_{stamps} = \frac{1}{1 + 58.04} \][/tex]
[tex]\[ P_{stamps} = \frac{1}{59.04} \][/tex]
- The probability, therefore, is approximately:
[tex]\[ P_{stamps} = 0.01693766937669377 \][/tex]
Therefore, the probabilities for the events are as follows:
- The probability of a random lottery ticket will win the jackpot is [tex]\(\displaystyle \frac{1}{195,130,343} \approx 5.124779594119813 \times 10^{-9} \)[/tex].
- The probability of an adult being struck by lightning during a year is [tex]\(\displaystyle \frac{1}{824,474} \approx 1.2128945242663807 \times 10^{-6} \)[/tex].
- The probability of an adult filing for personal bankruptcy during a year is [tex]\(\displaystyle \frac{1}{156.6} \approx 0.006385696040868455 \)[/tex].
- The probability of a person collecting stamps is [tex]\(\displaystyle \frac{1}{59.04} \approx 0.01693766937669377 \)[/tex].
