Q: From a group of 8 volunteers, including Andrew and Karen, 4 people are to be selected at random to organize a charity event. What is the probability that Andrew will be among the 4 volunteers selected and Karen will not?
I understand that combinatorics should be used to work out the number of ways you can get to 4 groups (70) but I then get a bit stumped. My assumption was to work out the likelihood of ANDREW being selected (which was a half) - this would therefore mean Karen being selected would be a half two in which case the chance of Andrew being selected and Karen not being selected is 1/2 * 1/2 = 1/4 however that does not get me to the correct answer, which is 2/7. Would anyone mind explaining how you work this one out?
Thanks!
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- AlexandraO504
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- Sage Pearce-Higgins
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Re: PROBABILITY PROBLEM SOLVING QUESTION
Please also post the full problem and source for this one.
You're right that Andrew will appear in exactly half of the groups, and so will Karen. However, your answer of 1/4 is very suspicious. First of all, that would render the information about 8 people irrelevant (you haven't used it), and 1/4 of 70 is 17.5, which is not a possible number of groups.
The reason is that if Andrew is selected, then there are 3 empty "slots" in the group, but 4 empty "slots" outside the group. So it's actually more likely that Karen will not be in the group if Andrew is selected. Let's consider the 35 possible groups of which Andrew is a member. We choose 3 members from the remaining 7 (that's what makes the 35 possible groups). The chance of having Karen chosen is 3/7, so the chance of her not being chosen is 4/7. So she'll be part of 15 groups and not part of 20 groups. So, there are 20 groups of which Andrew is a member but not Karen out of a total of 70, making 20/70 = 2/7.
My assumption was to work out the likelihood of ANDREW being selected (which was a half)
You're right that Andrew will appear in exactly half of the groups, and so will Karen. However, your answer of 1/4 is very suspicious. First of all, that would render the information about 8 people irrelevant (you haven't used it), and 1/4 of 70 is 17.5, which is not a possible number of groups.
The reason is that if Andrew is selected, then there are 3 empty "slots" in the group, but 4 empty "slots" outside the group. So it's actually more likely that Karen will not be in the group if Andrew is selected. Let's consider the 35 possible groups of which Andrew is a member. We choose 3 members from the remaining 7 (that's what makes the 35 possible groups). The chance of having Karen chosen is 3/7, so the chance of her not being chosen is 4/7. So she'll be part of 15 groups and not part of 20 groups. So, there are 20 groups of which Andrew is a member but not Karen out of a total of 70, making 20/70 = 2/7.
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