Conditional and Marginal Probabilites

[DoubtfulTurnip]

Show that the conditional probability of a survivor being male and the conditional probability
of a survivor being female are approximately a half.

	Dead	Survivor
male	387	75
female	89	76

I'm unsure on how to get an approximately a half here. What I've done is taken the total amount of survivors and the male survivors, then divided them. I did the same for the females. With those answers, I added them together and divided by 2. Which had resulted in a half, but I'm certain that's not the right way to go about it.

 

[BigBeachBanana]

Can you please share your work and possibility post a picture of the original question?

 

[DoubtfulTurnip]

Can you please share your work and possibility post a picture of the original question?

 

[JeffM]

What you have not done is to compute the margins explicitly. (If you are used to economics, "margin" in this case has nothing to do with the economic meaning of "margin." Here it is meant quite literally)

Please revise your table to compute the margins. Then let's proceed.

 

[DoubtfulTurnip]

Unsure what you mean here. I did originally divide everything by the total before to have all the numbers to equal to 1 originally, but found it troublesome to write it out. Should just use a different approach to my Bayes theorem. Like move the figures around ? Not entirely sure how else to make this table

 

[BigBeachBanana]

View attachment 31160View attachment 31158

Thank you for showing your work.
The first issue here is with your formula for the conditional probability.
Note that $\Pr(A|B)=\frac{\Pr(A \cap B)}{\Pr(B)}$
Second, Notice you have your events swapped.
The conditional probability of a survivor being male:
$$\Pr(\text{Male}|\text{Yes})=\frac{\Pr(\text{Male}\cap \text{Yes})}{\Pr(\text{Yes})}=\frac{75}{75+76}$$Similarly, The conditional probability of a survivor being female:
$$\Pr(\text{Female}|\text{Yes})=\frac{\Pr(\text{Female}\cap \text{Yes})}{\Pr(\text{Yes})}=\frac{76}{75+76}$$Convert $\frac{75}{151}$ and $\frac{76}{151}$ into decimal. You can see that they're both approximately 0.5.

 

[JeffM]

Ahh I did not see your work when I wrote my first repsonse. You did calculate the totals along the "margins" of your table.

Now this does not even work arithmetically.

$$\dfrac{\dfrac{75}{151} + \dfrac{76}{151}}{\dfrac{1}{2}} = \left ( \dfrac{75}{151} + \dfrac{76}{151} \right ) * \dfrac{2}{1} = \dfrac{150}{151} + \dfrac{152}{151} \approx 1 + 1 = 2.$$
Use your table. The number of males who survived is 75. The total who survived 151. Thus, the conditional probability of being male given being a survivoris simply
$$\dfrac{75}{151} \approx \dfrac{1}{2}.$$
The conditional probability of being female given being a survivor is simply

$$\dfrac{76}{151} \approx \dfrac{1}{2}.$$
The table makes it easy to compute conditional probabilities. Each cell is an "and" count. The totals on the bottom and to the right (on the margins of the main table) give you what to divide by to compute conditional probabilities..[/math]

 

[DoubtfulTurnip]

Thank you for your help, both of you @JeffM and @BigBeachBanana