# Probability of 1 Letter E in 4 Randomly Selected Letters from 'ENCYCLOPAEDIA

• Darkmisc
In summary, the probability of one letter E occurring in a random selection of four letters from the word ENCYCLOPAEDIA is 0.46. This can be calculated by dividing the number of ways to select one E and three non-E letters by the total number of ways to select four letters (13C4). Another method is to calculate the probability of getting an E as the first letter, followed by three non-E letters, and then multiplying by the number of possible combinations (4) for the placement of the E. The answer of 2/33 is incorrect.
Darkmisc
Homework Statement
Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one letter E will occur in the selection of 4 letters.
Relevant Equations
[SUP]2[/SUP]C[SUB]1[/SUB] x [SUP]11[/SUP]C[SUB]3[/SUB]/[SUP]13[/SUP]C[SUB]4[/SUB] = 0.46
Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one letter E will occur in the selection of 4 letters.

My calculation was

2C1 x 11C3/13C4 = 0.46

However, the answer is 2/33. I think the repeated letters might need to be accounted for. I divided the E term in the numerator by 2. I also divided the term for the remaining 11 letters by 4. I divided the denominator by 8 (2!x2!x2!), but this just canceled out the 2x4 in the numerator.
Thanks

Last edited:
I agree with your answer. I did it using direct probabilies. First, the probability of getting an ##E## as the first letter, followed by three letters that are not ##E## is:
$$P(EXXX) = \frac 2{13} \cdot \frac{11}{12} \cdot \frac{10}{11} \cdot \frac{9}{10} = \frac 3 {26}$$
Then we have the same probability for ##XEXX## etc. So, the total probability of precisely one ##E## is ##\frac{12}{26} = 0.46##.

Your method worked because you effectively labelled the duplicate letters ##E_1, E_2## and looked for either ##E_1## or ##E_2##. And the same with the other duplicate letters.

So, we have three ways to calculate and, happily, they all give the same answer.

PS you can see that ##2/33## can't be right, because you have only three options: no E's, one E ort two E's. The last of these looks the least likely, so you'd expect quite a high probability of either no E's or one E.

You could, as an exercise, calculate those probabilities as well: no E's and two E's. And check it all adds up to 1.

Last edited:
Darkmisc

## What is the probability of selecting at least 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA'?

The probability of selecting at least 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA' can be calculated using the formula P = 1 - (nC0 * (26-n)C4) / 26C4, where n is the number of letters in 'ENCYCLOPAEDIA' that are not E. In this case, n = 9, so the probability is 1 - (9C0 * 17C4) / 26C4 = 0.904.

## What is the probability of selecting exactly 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA'?

The probability of selecting exactly 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA' can be calculated using the formula P = (nC1 * (26-n)C3) / 26C4, where n is the number of letters in 'ENCYCLOPAEDIA' that are not E. In this case, n = 9, so the probability is (9C1 * 17C3) / 26C4 = 0.474.

## What is the probability of selecting no letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA'?

The probability of selecting no letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA' can be calculated using the formula P = (26-n)C4 / 26C4, where n is the number of letters in 'ENCYCLOPAEDIA' that are not E. In this case, n = 9, so the probability is 17C4 / 26C4 = 0.096.

## What is the probability of selecting all letter E's in 4 randomly selected letters from 'ENCYCLOPAEDIA'?

The probability of selecting all letter E's in 4 randomly selected letters from 'ENCYCLOPAEDIA' can be calculated using the formula P = nC4 / 26C4, where n is the number of letters in 'ENCYCLOPAEDIA' that are E. In this case, n = 3, so the probability is 3C4 / 26C4 = 0.

## What is the probability of selecting more than 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA'?

The probability of selecting more than 1 letter E in 4 randomly selected letters from 'ENCYCLOPAEDIA' can be calculated using the formula P = 1 - (nC0 * (26-n)C4) / 26C4, where n is the number of letters in 'ENCYCLOPAEDIA' that are not E. In this case, n = 9, so the probability is 1 - (9C0 * 17C4) / 26C4 = 0.904.

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