Combinatorial Probability

• Oct 11th 2010, 11:18 AM
NOX Andrew
Combinatorial Probability
There are two exercises in my probability and statistics textbook with which I am experiencing difficulty.

Exercise 1-28 A high school lottery uses two sets of numbered balls. One set consists of ten white balls numbered 1-10 and the second set contains twenty blue balls numbered 1-20. To play, you select two white balls and two blue balls.

(b) Your lottery ticket consists of four numbers: two white numbers, each between 1 and 10 inclusive, and two blue numbers, each between 1 and 20, inclusive. What is the probability that your lottery ticket contains exactly one matching white number and two matching blue numbers?

I know the total number of outcomes is $10^2 \times 20^2 = 40000$. How would I calculate the number of outcomes resulting in exactly one matching white number and two matching blue numbers (so I can divide that by the number of total outcomes to find the probability)?

Exercise 1-29 At a picnic, there was a bowl of chocolate candy that had 10 pieces each of Milky Way, Almond Joy, Butterfinger, Nestle Crunch, Snickers, and Kit Kat. Jen grabbed six pieces at random from this bowl of 60 chocolate candies.

(a) What is the probability that she got one of each variety?

There are $60 \cdot 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55 = \ _{60}P_6$ total outcomes. Of those outcomes, $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \ _6P_6$ result in one of each variety. Therefore, shouldn't the solution be $\frac{_6P_6}{_{60}P_6} = 1.99745 \cdot 10^{-8}$? The solution in the back of book is 0.01997.
• Oct 11th 2010, 07:55 PM
Soroban
Hello, NOX Andrew!

You should use Combinations, not Permutations.

Quote:

1-28. A high school lottery uses two sets of numbered balls.
One set consists of ten white balls numbered 1-10
and the second set contains twenty blue balls numbered 1-20.
To play, you select two white balls and two blue balls.

What is the probability that your lottery ticket contains exactly
one matching white number and two matching blue numbers?

There are: . ${10\choose2}{20\choose2} \:=\:45\cdot190 \:=\:8550$ possible outcomes.

We want one of the two White winners: 2 ways
[color=beige]and both of the two Blue winners: 1 way

Hence, there are 2 ways to have one white winner and two blue winners.

$P(\text{1 White, 2 Blue}) \;=\;\dfrac{2}{8550} \;=\;\dfrac{1}{4275}$

Quote:

1-29) At a picnic, there was a bowl of chocolate candy that had:
10 pieces each of Milky Way, Almond Joy, Butterfinger, Nestle Crunch,
Snickers, and Kit Kat.
Jen grabbed six pieces at random from this bowl of 60 chocolate candies.

What is the probability that she got one of each variety?

There are: . $\displaystyle {60\choose6} \:=\:50,\!630,\!860$ possible outcomes.

To get one of each, there are: . ${10\choose1}{10\choose1}{10\choose1}{10\choose1}{1 0\choose1}{10\choose1} \:=\:1,\!000,\!000$ ways.

$P(\text{one of each}) \;=\;\dfrac{1,\!000,\!000}{50,\!063,\!860} \;=\;\dfrac{50,\!000}{2,503,\!193}$