SOLUTION: Brenda has a jar that contains 5 blue tacks and 7 red tacks. She draws 3 tacks at random without replacement. What is the probability that all 3 are red?
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Question 1202343: Brenda has a jar that contains 5 blue tacks and 7 red tacks. She draws 3 tacks at random without replacement. What is the probability that all 3 are red? Found 2 solutions by math_tutor2020, ikleyn:Answer by math_tutor2020(3817) (Show Source):
I'll replace "tacks" with "marbles" since they are more friendly to fingers that reach in blindly to select at random.
Also, marbles are typically used in problems such as these.
Let's count the number of ways to select 3 red marbles.
We have n = 7 red marbles and r = 3 selections.
Use the nCr combination formula.
This formula is used because order doesn't matter.
n C r = (n!)/(r!(n-r)!)
7 C 3 = (7!)/(3!*(7-3)!)
7 C 3 = (7!)/(3!*4!)
7 C 3 = (7*6*5*4!)/(3!*4!)
7 C 3 = (7*6*5)/(3!)
7 C 3 = (7*6*5)/(3*2*1)
7 C 3 = 210/6
7 C 3 = 35
The value 35 can be found in Pascal's Triangle.
Look at the row that starts with 1,7,...
Count 4 spaces to the right to arrive at 35.
We count four spaces (instead of three) because the start index is r = 0.
There are 35 ways to select all three red marbles, from a candidate pool of seven red overall.
We now need to find how many ways there are to select any three marbles red and/or blue.
n = 5 blue + 7 red = 12 total
r = 3 selections
n C r = (n!)/(r!(n-r)!)
12 C 3 = (12!)/(3!*(12-3)!)
12 C 3 = (12!)/(3!*9!)
12 C 3 = (12*11*10*9!)/(3!*9!)
12 C 3 = (12*11*10)/(3!)
12 C 3 = (12*11*10)/(3*2*1)
12 C 3 = 1320/6
12 C 3 = 220
This value can also be found in Pascal's Triangle.
We found there are...
35 ways to get 3 red marbles.
220 ways to select any 3 marbles (red and/or blue).
Divide those values
35/220 = (5*7)/(5*44) = 7/44
7/44 = 0.1590909 = 15.90909% approximately
The "90" portion repeats forever.
Therefore, Brenda has about a 16% chance of getting all red marbles.
Another way to solve the problem is to calculate the total number of the tackes first, 5 + 7 = 12,
and then to write an expression for the probability
P = = reduce the fractions and get the answer = = . ANSWER
The formula is SELF-EXPLANATORY and, therefore, does not requires explanations.
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