The letters in the word MATHEMATICS are arranged randomly.
What is the probability that the first letter is E?
What is the probability that the first letter is M?
number of possible arrangements
= 11!/(2!2!2!) , 2 A's, 2 T's, 2 M's
= 4989600
number with E first
= 1x10!/(2!2!2!) = 453600
prob E first = 453600/4989600
= 4536/49896
= 1/11
with M first
number of arrangements
= 2x10!/(2!2!2!) = 907200
prob with M first = 907200/4989600
= 9072/49896
= 2/11
Ms Sue had both of those clearly using a very common sense reasoning
E - 1/11
M - 2/11
To calculate the probabilities, we need to know the total number of possible arrangements of the letters in the word "MATHEMATICS."
The word "MATHEMATICS" has a total of 11 letters. So, there are 11 different positions where we can place the first letter.
Let's go step-by-step:
1. Probability that the first letter is E:
The word "MATHEMATICS" has a total of 2 E's. Since we are interested in the probability of the first letter being E, we need to consider only one of the E's.
Number of favorable outcomes (where the first letter is E): 1
Total number of possible outcomes: 11
Therefore, the probability that the first letter is E is: 1/11.
2. Probability that the first letter is M:
The word "MATHEMATICS" has a total of 2 M's. We need to consider only one of the M's.
Number of favorable outcomes (where the first letter is M): 1
Total number of possible outcomes: 11
Therefore, the probability that the first letter is M is: 1/11.
So, the probability that the first letter is E is 1/11, and the probability that the first letter is M is also 1/11.
To determine the probabilities, we need to first count the total number of possible outcomes and the number of favorable outcomes for each case.
1. Probability that the first letter is E:
To find this probability, we count the number of favorable outcomes where the first letter is E, and divide it by the total number of possible outcomes.
Total number of possible outcomes:
The word "MATHEMATICS" has 11 letters, so there are 11 possible choices for the first letter.
Number of favorable outcomes:
Since we are given that the letters are arranged randomly, there is only 1 way for the first letter to be E.
Therefore,
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 11
So, the probability that the first letter is E is 1/11.
2. Probability that the first letter is M:
Using the same process as above:
Total number of possible outcomes: 11 (same as before)
Number of favorable outcomes: Similarly, there is only 1 way for the first letter to be M.
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 11
Thus, the probability that the first letter is M is also 1/11.