A STATE LOTTERY IS DESIGNED SO THAT A PLAYER CHOOSES FIVE NUMBERS FROM 1 TO 30 ON ONE LOTTERY TICKET. WHAT IS THE PROBABILITY THAT A PLAYER WITH ONE TICKET WILL WIN? WHAT IS THE PROBABILITY OF WINNING IF 100 DIFFERENT TICKETS ARE PURCHASED?
Number of possible combinations of 5 (distinct) numbers out of 30
= 30C5
Probability of winning if n different tickets are bought
= n/(30C5)
where 30C5
=30!/((30-5)!5!)
To calculate the probability of winning the lottery, we need to know the total number of possible outcomes and the number of favorable outcomes.
In this case, the total number of possible outcomes is the total number of ways to choose 5 numbers from 1 to 30. This can be calculated using the combinations formula, which is given by:
nCr = n! / (r!(n-r)!)
Where n is the total number of items, and r is the number of items chosen at a time.
So, to calculate the total number of possible outcomes, we have:
nCr = 30! / (5!(30-5)!)
= 30! / (5!25!)
Similarly, the number of favorable outcomes is just 1, since there is only one winning combination.
Now let's calculate the probability of winning with one ticket:
Probability of winning with one ticket = Number of favorable outcomes / Total number of possible outcomes
Probability of winning with one ticket = 1 / (30! / (5!25!))
To calculate the probability of winning if 100 different tickets are purchased, we need to consider that each ticket is a separate and independent event. So, the probability of winning with any one ticket remains the same.
Therefore, the probability of winning with 100 different tickets is simply the probability of winning with one ticket multiplied by the number of tickets purchased:
Probability of winning with 100 different tickets = Probability of winning with one ticket * 100
Remember to substitute the values of the total number of possible outcomes and the number of favorable outcomes into the formulas to calculate the probabilities.