Questions: New York State's "Quick Draw" lottery moves right along. Players choose between 1 and 10 numbers from the range 1 to 80; 20 winning numbers are displayed on a screen every four minutes. If you choose just one number, your probability of winning is 20/80, or 0.25. Lester plays one number ten times as he sits in a bar. What is the probability that all ten bets lose? Give your answer to four decimal places.

Solution Steps

Lester plays the New York State "Quick Draw" lottery by choosing one number and placing ten bets. The probability of winning with one bet is given as \( p = \frac{20}{80} = 0.25 \). Consequently, the probability of losing a single bet is \( q = 1 - p = 0.75 \).

We are interested in the probability that all ten bets lose. This scenario can be modeled using a binomial distribution where:

• \( n = 10 \) (the number of trials),
• \( x = 0 \) (the number of successes, which in this case means winning bets).

The probability of exactly \( x \) successes in \( n \) trials is given by the formula: \[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Substituting the values into the formula, we have: \[ P(X = 0) = \binom{10}{0} \cdot (0.25)^0 \cdot (0.75)^{10} \] Calculating this gives: \[ P(X = 0) = 1 \cdot 1 \cdot (0.75)^{10} = (0.75)^{10} \] Calculating \( (0.75)^{10} \) yields approximately \( 0.0563 \).

Final Answer