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14 votes
14 votes
Find the probability of each event

A bag contains nine real diamonds and
three fake diamonds. If nine diamonds are
picked from the bag at random, what is the
probability that all of them are real?

User Metalshark
by
2.4k points

1 Answer

11 votes
11 votes

Answer: If replaced: 0.075 or 7.5%

If not replaced: 0.00455 or 0.455%

Explanation:

If the diamonds are replaced after each draw:

The probability of drawing a real diamond is 9/12, found by making the numerator the number of real diamonds and adding the total number of diamonds to find the denominator. 3+9 = 12.

9/12 can be reduced to 3/4 by dividing the numerator and denominator by 3.

To find the probability of all 9 diamonds drawn being real, multiply 3/4 by itself 9 times.
3^9 = 19683.
4^9 = 262144. This creates a probability of 19683/262144. This cannot be reduced because there is no common factor. To express as a decimal, divide. 19683/262144 = 0.07508. To turn this into a percent, multiply by 100. 0.07508*100 = 7.508

If the diamonds are not replaced after each draw:

Start with the probability of drawing a single diamond: 9/12. After each draw, both the numerator and denominator have 1 subtracted from them, since there is 1 less real diamond and 1 less diamond in total. This will give you the equation (9/12)*(8/11)*(7/10)*(6/9)*(5/8)*(4/7)*(3/6)*(2/5)*(1/4). This equals 0.00455. To find as a percent, multiply by 100. 0.00455*100 = 0.455

User Exceen
by
3.3k points