Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets at least $6$ chocolates?
First, let's find the total number of ways Miyu and distribute the chocolates with no restrictions.
According to the Stars and Bars Theorem, we have
where n is the identical items and k is the number of groups.
Plugging in 8 and 5, we have
Next, we must find the number of favorable outcomes. If Druv gets 6 chocolates, then there are 2 more chocolates for 4 other friends.
Using the same method, we have
Now, we can write the equation
So our answer is approximately 2.02%.
Thanks! :)
In that case, we can do the same with 7 and 8.
We have
and
So there are 10+4+1 = 15.
So our answer is 15/495.
Thanks! :)