n coins are flipped, and you are given that at least 1 of them is heads. The probability that at least one other coin is heads is 19/21. How many coins do you have?
Hello, mark1950!
An unusual problem . . .
I had to come up with an unusual solution.
When coins are flipped, there are possible outcomes.coins are flipped, and you are given that at least 1 of them is heads.
The probability that at least one other coin is heads is
How many coins do you have?
We are told that at least one of the coins was Heads.
. . Then there are: . outcomes.
We eliminate the case of "no Heads" (all Tails).
We want the probability that at least two coins are Heads.
. . There is 1 outcomes with 0 Heads.
. . There are outcomes with exactly 1 Head.
Hence, there are: . outcomes with at least 2 Heads.
The probability of at least two Heads, given there is at least one Head is
. .
. .
There is no elementary method for solving this equation.
By inspection, I found the solution: .
Erm, I don't really get this step. If you don't mind, can you please explain in a more specific detail? Thanks.
We want the probability that at least two coins are Heads.
. . There is 1 outcomes with 0 Heads.
. . There are n outcomes with exactly 1 Head.
Hence, there are: .2^n - n - 1 outcomes with at least 2 Heads.
And also, how do you do inspections to get n = 6?
If Soroban doesn't mind my jumping in here:
With n coins there are possible outcomes: each coin can come up either heads or tails so each coin multiplies the possiblities by 1. Exactly one of those has "all tails" so have "at least one head.
There are exactly n outcomes with exactly one head because there are n positions that "head" can be in. To find the number of possible outcomes with 2 or more heads, start with total and subtract off the number with one head or no heads: . The probability that "given that there is at least one head, there are at least 2" is and that must be 19/21.
Now just calculate a few values: if n= 2 (there must be at least two flips to get two heads!), .
If n= 3, .
If n= 4,
If n= 5,
if n= 6, !