The letters in the word MADAGASCAR are placed in a bag. Three letters
are taken out at random without replacement. What is the probability that
the letters A, R and C are drawn, in that order?
The probability of drawing A, R and C in that order is 1/20. This is because there are a total of 10 letters in MADAGASCAR, and the probability of drawing any one letter is 1/10. Since the letters are drawn without replacement, the probability of drawing A, R and C in that order is 1/10 x 1/9 x 1/8 = 1/720.
Another reply by the bot which is WRONG, and a disservice to students
prob(A, then R, then C) = (4/10)(1/9)(1/8) = 1/180
To find the probability, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Total number of outcomes:
There are 10 letters in the word MADAGASCAR.
The first letter can be any of the 10 letters.
The second letter can be any of the remaining 9 letters.
The third letter can be any of the remaining 8 letters.
Total outcomes = 10 * 9 * 8 = 720
Favorable outcomes:
Since we want the letters A, R, and C to be drawn in that order, we need to consider the following possibilities:
1. The first letter is A, the second letter is R, and the third letter is C.
2. The first letter is A, the second letter is C, and the third letter is R.
For each case, the probability of drawing the specific letter is:
Case 1:
The first letter can only be A.
The second letter can only be R.
The third letter can only be C.
Case 2:
The first letter can only be A.
The second letter can only be C.
The third letter can only be R.
Therefore, the number of favorable outcomes is 1 + 1 = 2.
Probability = Favorable outcomes / Total outcomes
Probability = 2 / 720
Simplifying the fraction, we get:
Probability = 1 / 360.
Therefore, the probability that the letters A, R, and C are drawn in that order is 1/360.
To find the probability of drawing the letters A, R, and C in that order from the word "MADAGASCAR," we need to first determine the total number of possible outcomes and then determine the number of desired outcomes.
Step 1: Total number of possible outcomes
The word "MADAGASCAR" has a total of 11 letters. When we draw three letters at random without replacement, the number of possible outcomes can be calculated using combinations. The formula for combinations is nCr = n! / (r!(n-r)!), where n represents the total number of items and r represents the number of items chosen at a time.
In this case, we have 11 items (letters) and we are choosing 3 at a time. Therefore, the total number of possible outcomes is calculated as follows:
Total possible outcomes = 11 C 3 = 11! / (3!(11-3)!) = 11! / (3! 8!) = (11 * 10 * 9) / (3 * 2 * 1) = 165
Step 2: Number of desired outcomes
In order for the letters A, R, and C to be drawn in that specific order, the first letter drawn must be A, the second must be R, and the third must be C.
The probability of drawing any specific letter from the bag is 1/11 for the first draw, as there are 11 letters in total and only 1 A. Similarly, the probability of drawing R as the second letter is 1/10, as there are now 10 letters remaining in the bag and only 1 R. Finally, the probability of drawing C as the third letter is 1/9, as there are 9 letters remaining and only 1 C.
To determine the number of desired outcomes, we multiply the probabilities of each letter being drawn in order:
Number of desired outcomes = (1/11) * (1/10) * (1/9) = 1/990
Step 3: Finding the probability
Finally, we can calculate the probability by dividing the number of desired outcomes by the total possible outcomes:
Probability = Number of desired outcomes / Total possible outcomes
Probability = (1/990) / (165/1) = 1/990 * 1/165 = 1/163,350
Therefore, the probability of drawing the letters A, R, and C in that specific order from the word "MADAGASCAR" is 1/163,350.