A certain state's lottery game, LOTTO, is set up so that each player pays $1 per ticket and chooses six different numbers between 1 and 20 inclusive.(the order doesn't matter) If the six numbers chosen match the six numbers drawn, the player wins the grand prize
If a person buys 1000 LOTTO tickets, what is his probability of winning the grand prize?
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To calculate the probability of winning the grand prize in the LOTTO game, we need to consider the total number of possible outcomes and the number of favorable outcomes.
First, let's determine the total number of possible outcomes. Each ticket allows the player to choose 6 numbers from a pool of 20 numbers. Since the order doesn't matter, we can use the combination formula. The formula for combinations is given by:
nCr = n! / r!(n-r)!
Here, n represents the total number of items, and r represents the number of items being chosen.
In this case, we have 20 numbers to choose from, and we need to choose 6 numbers, so:
n = 20 and r = 6
Plugging these values into the combination formula:
20C6 = 20! / 6!(20-6)!
= 20! / 6!14!
= (20 x 19 x 18 x 17 x 16 x 15) / (6 x 5 x 4 x 3 x 2 x 1)
= 38,760
Therefore, the total number of possible outcomes is 38,760.
Next, let's determine the number of favorable outcomes, which is 1 because there is only one winning combination.
By dividing the number of favorable outcomes by the total number of possible outcomes, we can find the probability of winning the grand prize:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 1 / 38,760
= 0.0000258
So, the probability of winning the grand prize with 1000 LOTTO tickets is 0.0000258 or approximately 0.00258%.