I'm looking at my notes and I can't seem to figure how my teacher did this problem. She used her name which has 9 letters, 3 vowels, and 6 consonants. If we were to reach in the bag and grab 3 letters out. By using the probability distribution what's the probability of getting 0, 1, 2, and 3 vowels. I know how to do the probability of 0 vowel. For 1 vowel she did (3/9*6/8*5/7)3 how did she get 6/8 and 5/7. For 2 vowels she did (3/9*2/8*6/7)3 same here where did the 2/8 and 6/7 come from. Then for 3 vowels she had 1 minus everything.
I will do the 2 vowels for you
since she grabbed 3 letters it could have been
VVC, VCV, or CVV
look at the first, the VVC
there are 3 vowels out of 9 prob (first a vowel) = 3/9
so now there are 8 letters left, 2 of them vowels
prob (second a vowel) = 2/8
so now the third letter must be a consonant, and there are 7 letters left, 6 of them consonants.
Prob(third is a consonant) = 6/7
so
Prob(first a vowel, 2nd a vowel, third a cons) = 3/9 x 2/8 x 6/7
now can you see how she would get the VCV of
3/9 x 6/8 x 2/7 setup ?
and for the CVV it would be 6/9 x 3/8 x 2/7
did you notice that the numerators are simply
2x3x6 and the denominators are 9x8x7 just in different order?
Does it matter what order you do multiplication?
so the prob would be
(3/9*2/8*6/7)(3/9*2/8*6/7)(3/9*2/8*6/7)3
=(3/9*2/8*6/7)^3
Now see if you reason out the others in the same way.
To understand how your teacher arrived at the probabilities for 1 vowel and 2 vowels, let's break it down step by step.
First, let's consider the probability of selecting 1 vowel out of the 3 letters drawn.
1. There are 3 vowels in your teacher's name, and since you are choosing 3 letters, the first step is to choose 1 vowel from the 3 available. The probability of selecting 1 vowel is therefore 3/9.
2. After selecting 1 vowel, you are left with 6 consonants out of the remaining 8 letters. So the probability of selecting 1 consonant is 6/8.
3. Finally, there are 5 remaining letters in the bag, out of which you need to select 1 letter. Therefore, the probability of selecting a second consonant is 5/7.
To find the total probability of getting 1 vowel, you need to multiply these individual probabilities together because you want all three events (choosing 1 vowel, then 1 consonant, then 1 consonant) to occur. Hence, the probability of getting 1 vowel is:
(3/9) * (6/8) * (5/7) = (3/9) * (3/4) * (5/7) = 45/252 = 5/28.
Now let's move on to finding the probability of getting 2 vowels.
1. The probability of selecting 2 vowels will be calculated in a similar manner. You choose 2 vowels from the 3 available, giving a probability of 3/9.
2. After selecting 2 vowels, you are left with 6 consonants out of the remaining 8 letters. So the probability of selecting 1 consonant is 6/8.
3. Finally, there are 7 remaining letters in the bag, out of which you need to select 1 letter. Therefore, the probability of selecting a third consonant is 7/7, which simplifies to 1.
To find the total probability of getting 2 vowels, you multiply these individual probabilities together because you want all three events (choosing 2 vowels, then 1 consonant, then 1 consonant) to occur. Hence, the probability of getting 2 vowels is:
(3/9) * (6/8) * (7/7) = (3/9) * (3/4) * 1 = 9/36 = 1/4.
For the probability of getting 3 vowels, your teacher used the method of finding the complement. The complement of getting 0 vowels, 1 vowel, and 2 vowels is getting 3 vowels, so it can be calculated as:
1 - (probability of getting 0 vowels + probability of getting 1 vowel + probability of getting 2 vowels).
I hope this explanation helps you understand the logic behind your teacher's calculations.