Algebra Please Explain

Question

You have 11 cards with the letters PROBABILITY written on them, one letter per card. Without replacing any cards, you randomly select two cards. Consider "Y" as a vowel. What is the probability of selecting a vowel and a consonant? Write your answer as a fraction.

1)
3/11


2)
6/11


3)
30/11


4)
16/30

Answer 1

Since it is two consecutive events in a row, you have to multiply the separate event probabilities together.

 

The first event is selecting a Vowel.  In the word PROBABILITY, there are 5 vowels, even though I is repeated, it still gives us another chance at a vowel and they also want you to count Y as a vowel.  Therefore the probability of choosing a vowel is 5/11.  The next event is a consonant.  The left over letters are consonants, so we have 6 consonants.  However, since we took one card out, we only have 10 cards to choose from, thus the probability of a consonant on our second pick is 6/10

 

So now we multiply these two fractions and simplify:

 

5/11 * 6/10 = 30/110 = 3/11

Answer 2

When you find the probability of two events happening in row, you multiply the probabilities of each event.  However, because you are not replacing the cards, this changes the probability of the second event because there is now now fewer cards.

 

P(vowel) = 5/11 (5 vowels out of the 11)

 

Normally, P(consonant) = 6/11 (6 consonants out of the 11), but since we already pulled the vowel, it becomes 6/10 since there is one fewer card.

 

Therefore, P(Vowel and Consonant) = 5/11 * 6/10= 30/110 = 3/11  (Option 1)