The cards bearing letters of the word "MATHEMATICS" are placed in a bag. A card is taken out from the bag without looking into the bag (at random).
(a) How many outcomes are possible when a letter is taken out of the bag at random?
(b) What is the probability of getting
(i) M?
(ii) Any vowel?
(iii) Any consonant?
(iv) X?

Answer

To solve the problem step by step, let's break it down according to the parts of the question.

### Step 1: Identify the total number of letters in "MATHEMATICS"
The word "MATHEMATICS" consists of the following letters:
- M, A, T, H, E, M, A, T, I, C, S

Now, let's count the occurrences of each letter:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1

### Total number of letters:
To find the total number of outcomes when a letter is taken out of the bag, we count all the letters:
- Total letters = 2 (M) + 2 (A) + 2 (T) + 1 (H) + 1 (E) + 1 (I) + 1 (C) + 1 (S) = 11

**Answer for part (a):** The total number of outcomes is **11**.

### Step 2: Calculate the probability of getting M
To find the probability of getting M:
- Number of favorable outcomes for M = 2 (since M appears twice)
- Total outcomes = 11

Probability of getting M = Number of favorable outcomes / Total outcomes = 2 / 11

**Answer for part (b)(i):** The probability of getting M is **2/11**.

### Step 3: Calculate the probability of getting any vowel
The vowels in "MATHEMATICS" are A, E, and I:
- A: 2
- E: 1
- I: 1

Total number of vowels = 2 (A) + 1 (E) + 1 (I) = 4

Probability of getting a vowel = Number of favorable outcomes for vowels / Total outcomes = 4 / 11

**Answer for part (b)(ii):** The probability of getting any vowel is **4/11**.

### Step 4: Calculate the probability of getting any consonant
The consonants in "MATHEMATICS" are M, T, H, C, and S:
- M: 2
- T: 2
- H: 1
- C: 1
- S: 1

Total number of consonants = 2 (M) + 2 (T) + 1 (H) + 1 (C) + 1 (S) = 7

Probability of getting a consonant = Number of favorable outcomes for consonants / Total outcomes = 7 / 11

**Answer for part (b)(iii):** The probability of getting any consonant is **7/11**.

### Step 5: Calculate the probability of getting X
The letter X does not appear in "MATHEMATICS":
- Number of favorable outcomes for X = 0

Probability of getting X = Number of favorable outcomes for X / Total outcomes = 0 / 11 = 0

**Answer for part (b)(iv):** The probability of getting X is **0**.

### Summary of Answers:
(a) Total outcomes = **11**
(b)(i) Probability of M = **2/11**
(b)(ii) Probability of any vowel = **4/11**
(b)(iii) Probability of any consonant = **7/11**
(b)(iv) Probability of X = **0**