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Two marbles are drawn, without replacement, from an urn containing 4 red marbles, 5 white marbles, and 2 blue marbles. Determine the probability that:

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3 Answers

Probability is the # of possible outcomes desired /  # of possible total outcomes.  In problems like this when you have consecutive draws, you find the probability of each individual draw and then multiply your results.
a. there are 4 reds to possible draw from the urn, there are 11 total marbles.  The probability of drawing 1 red is 4/11. 
After the 4/11 chance that you did get a red, now what is the probability of getting a second red? 
*Note, the was no replacing so you have 1 red in your hand and 3 in the urn. 
Now there is a 3/10 probability of getting the second red marble. 10 because you have one of the 11 in your hand so there are only 10 left in the urn. 
so we have 4/11 and 3/10, multiply and we get 12/110 which reduces to 6 /55
b. same idea just keep your colors and number of marbles straight. 
P(red) = 4/11
P(b after a red has been drawn) = 2/10, 2 blue still in the urn but only 10 marbles since you have 1 red in your hand.
Multiply 4/11 and 2/10 which equals 8/110 = 4/55
c. P(b) = 2/11, P(red after a blue has been drawn) = 4/10,  multiply 8/110 = 4/55
d. this one is different, you have to account for either one being red or blue.  so a red then blur or a blue then a red.  We have already found these probabilities in b. and c.  Since both situations need to be counted then we add them together.  4/55 +4/55 = 8/55
Why don't we multiply?  Because that would give us a smaller probability and if you think about the problem, out chances increase if the order doesn't matter. (yes there is a mathematical explanation but I like  to use common sense when possible)
Remember: P(A and B)=P(A)P(B) for independent events (which marble drawings are)
P(A or B)=P(A)+P(B) for disjoint events
1. P(red and red) =(4/11)(3/10)=12/110
2. P(red and blue)=(4/11)(2/10)=8/110
3. P(blue and red)=(2/11)(4/10)=8/110
4. P(red or blue)=P(red and blue)+P(blue and red)=16/110