I am thinking of a four digit number. The sum of the four digits is 6, but none of the digits is zero. If you have only one opportunity to guess the number, what is the probability that your guess is correct?
Obviously, digits must be allowed to repeat, otherwise
the smallest combination would be 1234, and that sum is 10
So allowable choices:
1113 ---> arrange in 4!/3! or 4 ways
1122 ---> arrange in 4!/(2!2!) = 6 ways
so there are 10 such numbers.
So if you guess following the restriction, it would be a prob of 1/10
Thx!
To find the probability of guessing the correct number, we need to determine the total number of possible four-digit numbers that meet the given conditions, and then find the probability of randomly guessing the correct number among those possibilities.
1. Finding the total number of possibilities:
Since none of the digits is zero, we have 9 choices for each digit (i.e., they can take on values between 1 and 9, inclusive). The sum of the four digits is 6, so we need to divide the sum of 6 among the four digits. We can think of this as distributing 6 identical balls (representing the sum) among 4 distinct boxes (representing the digits).
Using the concept of stars and bars (also known as combinations or compositions), this can be represented as finding the number of ways to arrange 6 identical balls and 3 dividers among the four boxes.
The total number of possibilities is given by the formula (6 + 3) C 3 = 9 C 3 = 9! / (3! * (9-3)!).
2. Finding the probability of guessing the correct number:
Since we only have one opportunity to guess the number, there is only one correct guess out of the total possibilities.
Therefore, the probability of guessing the correct number is 1 divided by the total number of possibilities calculated in step 1.
Now, let's calculate the probability:
Total number of possibilities = 9 C 3 = 9! / (3! * (9-3)!).
Simplifying, we get 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84.
Probability of guessing the correct number = 1 / (Total number of possibilities) = 1 / 84.
Hence, the probability of guessing the correct number is 1/84.