Consider a large bag of coins, consisting of only quarters ($0.25), dimes ($0.10), and nickels ($0.05). 40% of the coins are nickels, 35% are dimes, and 25% are quarters.
You must draw either 5 quarters of 4 quarters and a nickel or dime.
Suppose there are 100N coins consisting of 40N nickels, 35N dimes, and 25N
quarters. Note: there are 75N non-quarters
The number of successes equals
the number of ways to draw 5 quarters is C(25N,5)
plus
the number of ways to draw 4 quarters and 1 non-quarter is C(25N,4)·(75N)
So the number of successes is
C(25N,5) + C(25N,4)·(75N)
+
Put the (75N) in the numerator of the second fraction
+
We get a least common denominator of 5! by multiplying the second fraction
by
+
+
Factor the numerator:
The number of possible ways is the number of ways to select any
5 coins from the 100N coins:
C(100N,5) =
So the probability is
÷
×
×
The first fraction is , exactly .
If N is large the next three are also very nearly
or , and the last fraction is very nearly , or 4.
So that is very nearly
That's not exact, but it is very close if N is large.
Edwin