Difference between revisions of "2014 AMC 8 Problems/Problem 18"
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So out of the four fractions, D is the largest. So our answer is <math>\boxed{\text{(D)}}.</math> | So out of the four fractions, D is the largest. So our answer is <math>\boxed{\text{(D)}}.</math> | ||
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We can also find out how many total cases there are for one solution. This will work, because before simplifying, the denominators of the fraction will be the same. Both <math>\text{A}</math> and <math>\text{B}</math> have <math>{4 \choose 0}=1</math> possibility. <math>\text{C}</math> will have <math>{4 \choose 2}=6</math> possibilities. <math>\text{D}</math> has <math>2\cdot{4 \choose 1}=8</math> possibilities (note that the problem did not say a specific gender.) Therefore, <math>\boxed{\left(\text{D}\right)\text{ 3 are of one gender and 1 is of the other gender}}</math> will have the greatest probability of occurring. | We can also find out how many total cases there are for one solution. This will work, because before simplifying, the denominators of the fraction will be the same. Both <math>\text{A}</math> and <math>\text{B}</math> have <math>{4 \choose 0}=1</math> possibility. <math>\text{C}</math> will have <math>{4 \choose 2}=6</math> possibilities. <math>\text{D}</math> has <math>2\cdot{4 \choose 1}=8</math> possibilities (note that the problem did not say a specific gender.) Therefore, <math>\boxed{\left(\text{D}\right)\text{ 3 are of one gender and 1 is of the other gender}}</math> will have the greatest probability of occurring. | ||
Revision as of 18:46, 26 December 2021
Contents
Problem
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
Solution 1
We'll just start by breaking cases down. The probability of A occurring is 
. The probability of B occurring is .
The probability of C occurring is 
, because we need to choose 2 of the 4 slots to be girls.
For D, there are two possible cases, 3 girls and 1 boy or 3 boys and 1 girl. The probability of the first case is 
because we need to choose 1 of the 4 slots to be a boy. However, the second case has the same probability because we are choosing 1 of the 4 children to be a girl, so the total probability is 
.
So out of the four fractions, D is the largest. So our answer is 
Solution 2
We can also find out how many total cases there are for one solution. This will work, because before simplifying, the denominators of the fraction will be the same. Both 
and 
have 
possibility. 
will have 
possibilities. 
has 
possibilities (note that the problem did not say a specific gender.) Therefore, 
will have the greatest probability of occurring.
See Also
| 2014 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
