Question 200135
a)

Let's build the tree:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/work/w5a39a-2.png">


Using the tree, we get the sample space:


{b,b,b,b}
{b,b,b,g}
{b,b,g,b}
{b,b,g,g}
{b,g,b,b}
{b,g,b,g}
{b,g,g,b}
{b,g,g,g}


{g,b,b,b}
{g,b,b,g}
{g,b,g,b}
{g,b,g,g}
{g,g,b,b}
{g,g,b,g}
{g,g,g,b}
{g,g,g,g}


So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order).


Note: recall, the sample space is the set of ALL possible outcomes.

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b)

Since we want to know the chances of the couple having "at least three boys", this means they want to know the chances of having 3 boys OR 4 boys (since <i>at least</i> means that figure or more).



Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys:

{b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b}


So there are 4 cases where the couple would have 3 boys. 



So *[Tex \LARGE P(\text{3 boys ONLY})=\frac{\text{Number of Cases with 3 boys only}}{\text{Total number in sample space}}=\frac{4}{16}=\frac{1}{4}] 


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Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that
 
 
 
*[Tex \LARGE P(\text{\text{4 boys ONLY}})=\frac{\text{Number of Cases with 4 boys only}}{\text{Total number in sample space}}=\frac{1}{16}]



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Now simply add the two probabilities to find the chances of either one occurring:


*[Tex \LARGE P(\text{AT LEAST 3 boys})=P(\text{3 boys OR 4 boys})]



*[Tex \LARGE P(\text{AT LEAST 3 boys})=P(\text{3 boys ONLY})+P(\text{4 boys ONLY})]



*[Tex \LARGE P(\text{AT LEAST 3 boys})=\frac{\text{Number of Cases with 3 boys only}}{\text{Total number in sample space}}+\frac{\text{Number of Cases with 4 boys only}}{\text{Total number in sample space}}]



*[Tex \LARGE P(\text{AT LEAST 3 boys})=\frac{4}{16}+\frac{1}{16}]



*[Tex \LARGE P(\text{AT LEAST 3 boys})=\frac{4+1}{16}]



*[Tex \LARGE P(\text{AT LEAST 3 boys})=\frac{5}{16}]



So the probability of the couple having AT LEAST 3 boys is {{{5/16}}} which is 0.3125 in decimal form which gives a 31.25% chance.