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Practice with Contingency Tables and Conditional Probability

Show your work or explain your thinking. There may be a few different methods that work.

1. According to the 2019 U.S. Census, the country's total population was 328,239,523 people. The Census only uses male and female genders: 50.8% of the population was female, and the rest male. The census does not ask about handedness, but other studies estimate 11.8% of males are left-handed, and 9.6% of females are left-handed. Use this information to fill out the table below.

For the follwing problems, use the following labels for situations: L = left handed, R = right handed, M = male, and F = female.

2. The statement "11.8% of males are left-handed" can be rephrased as the probability a person is left-handed, given that the person is male, is 11.8%. The notation for this conditional probability is P(L|M) and we could write the equation P(L|M) = 0.118. Write a similar equation for the statement "9.6% of females are left-handed".

3. Would you say that gender and handedness are independent? That is, does one’s gender affect one’s likelihood of being left handed or right handed?

4. Recall that for independent events A and B, P(A and B) = P(A) × P(B). Can we say that P(M and L) = P(M) × P(L)?

5. In general, if events A and B are not independent, then we write P(A and B) = P(A) × P(B|A). Knowing L and M are not independent, write P(M and L) as a product.

6. Knowing L and F are not independent, write P(F and L) as a product.

The DSM-5 estimates that 0.005% to 0.014% of male newborns will experience gender dysphoria at some time in their life. Studies have linked high levels of maternal first-trimester intrauterine testosterone to both left-handedness and gender dysphoria. Furthermore, a 2001 study found 19.5% of its boys with gender dysphoria were left-handed.

7. Use this information to finish filling out the table below for Oregon's population of 700,000 boys of age 14 years or younger, picking the high estimate of 0.014% and assuming that 11.8% of boys without gender dysphoria are left-handed.

Boys with Gender Dysphoria Boys without Gender Dysphoria Total Step 3 is 0.118 × 699,902 =82,588 Step 1 is 0.00014 × 700,000 =98 Step 2 is 700,000 − 98 =699,902 700,000

For the follwing problems, use the following labels for situations: L = left handed, R = right handed, N = boy without Gender Dysphoria, and Y = boy with Gender Dysphoria.

8. Find P(L|N).

9. Find P(N|L).

10. Find P(N|R).

11. Find P(Y|L).

12. Find P(Y|R).

13. Find P(Y|L) − P(Y|R). (This is an absolute change.)

14. Divide the answer to #13 by P(Y|R). (This is a relative change.)