Question

Suppose you play a "daily number" lottery game in which three digits from 0–9 are selected at random, so your probability of winning is 1/1000. Also suppose lottery results are independent from day to day.

A. If you play every day for a 7-day week, what is the probability that you lose every day?

B. If you play every day for a 7-day week, what is the probability that you win at least once? (Hint: Make use of your answer to part a.)

C. Repeat parts a and b if you play every day for a 30-day month.

D. What if each digit can only be selected once. How many different ways can the three digits be selected?

E. If order does matter in the digits, how many different ways can the numbers be arranged?

Answer 1

The probability of winning a lottery game per day is 1/1000. The lottery results are independent.

A) P(Lose every day)

The probability of lose at each day = 1 - probability of winning = 1 - (1/1000) = 999 / 1000

P(You lose every day for a 7 day week)  

=P(lose on first day) * P(lose on second day)*P(lose on third day)*P(lose on fourth day)*P(lose on fifth day)*P(lose on sixth day)*P(lose on seventh day)

= (999/1000)*(999/1000)*(999/1000)*(999/1000)*(999/1000)*(999/1000)*(999/1000)

= 0.9930

If you play everyday for a 7 day week, the probability that you lose everyday is 0.9930

B) P(You win at least once in 7 day week)

At least one means either 1, 2 , 3, 4, 5, 6 or 7.

Instead of finding all these probabilities, use compliment rule

P(At least one) = 1 - P(X = 0)

X = 0 means lose on every day that is lose on all 7 days in week

P(X = 0) = P(lose everyday) = 0.9930 (from A part)

P(win at least once in 7 day week) = 1 - 0.9930 = 0.0070

Therefore, the probability that you win at least once is 0.0070

C)

You play every 30 day of month

P(lose every 30 day of month) = P(lose on first day(=) * P(lose on second day)*....*P(lose on 30th day)

= (999/1000) * (999/1000) * ...*P(999 / 1000)

= (999/1000)³⁰ = 0.9704

Therefore, the probability that you lose everyday for 30 day month is 0.9704

P(win at least once) = 1 - P(lose on all 30 day) = 1 - 0.9704 = 0.0296

Therefore, the probability that you win at least once is 0.0296

D)

Digit can only be selected once that is there is no replacement. that is one selected digit will less for next choice

Total number of different ways = for first digit there are total 10 possibles ways as 0 to 9 * number of choices for second digit after selecting the first * number of choices for third digit after selecting the first two.

= 10 * 9 * 8 = 720

Therefore, there are 720 different ways can the three digits be selected.

E)

When order of selection matters then use permutation to select the possible ways.

Out of total 10 digits 3 need to selected so it is ¹⁰P₃

which is nothing but 10! / (10 - 3) = 10! / 7! = 3628800 / 5040 = 720

There are total 720 different ways can the numbers be arranged.