**Distributome Colorblindness Activity**

**Overview**

Colorblindness – Can you see the number in this image?

This Distributome Activity illustrates an application of probability theory to study Colorblindness, typically a genetic disorder which results from an abnormality on the X chromosome. The condition is thus rarer in women since a woman would need to have the abnormality on both of her X chromosomes in order to be colorblind (whether a woman has the abnormality on one X chromosome is essentially independent of having it on the other).

**Goals**

The goal of this activity is to demonstrate an efficient protocol of estimating the probability that a randomly chosen individual may be colorblind.

**Hands-on Activity**

Suppose that \(p\) is the probability that a randomly selected ”man” is colorblind.

- 100 men are selected at random. What is the distribution of \(X_m\) = the number of these men that are colorblind?

- 100 women are selected at random. What is the distribution of \(X_f\) = the number of these women that are colorblind?

- To estimate the probability that a randomly selected woman is colorblind, you might use the proportion of colorblind women in a sample of n women. What is the variance of this estimator?
- Alternatively, to estimate the probability that a randomly selected woman is colorblind, you might use the square of the proportion of colorblind men in a sample of n men. Explain why this estimate makes sense. What is the variance of this estimator?

- For large samples, is it better to use a sample of men or a sample of women to estimate the probability that a randomly selected women is colorblind? Explain.

**Alternate approach**

You can also use the delta method to find the approximate variance for the estimator above.

**Conclusions**

In practice, it may difficult to obtain reliable parameter estimates when the event at hand is very rare (such as with colorblindness in women). The use of a valid probability model such as the relationship between the chance of colorblindness in men and the chance in women may improve these estimates.