Question

It turns out that there is a strong relationship between gender, age and the likelihood of fatalities.  Consider the following two tables, Exhibits 4 and 5, that show proportions of fatalities from each category.

Exhibit 4

	17 to 25	26 to 64	65 or Older	Total
Female	6.3%	13.7%	4.6%	24.6%
Male	21.4%	42.0%	12.0%	75.4%
Total	37.0%	56.0%	17.0%	100%

Exhibit 5

	17 to 25	26 to 64	65 or Older	Total
Female	22.8%	24.6%	27.7%	24.6%
Male	77.2%	75.4%	72.3%	75.4%
Total	100%	100%	100%	100%

A) Take a randomly chosen fatality.

1. What is the probability that the person was male?  
2. What is the probability that the person was a female aged 65 or older?
3. If we know the person killed was aged 65 or older, what is the probability they are female?
4. What are the three types of probabilities used in each of these questions (i), (ii), and (iii)?

B) Consider the comment, “77.2% of fatalities amongst 17-25 year olds are males. This just shows how careless young male drivers are”.

1. Do the numbers in Exhibit 5 provide some evidence to support the conclusion outlined in the comment? Explain.
2. Suppose I tell you that there are many more male drivers in the age range 17-25 than there are female drivers in this range. Does this undermine the conclusion in the comment? Explain.

Answer 1

From the given proportion of fatalities:

i. The probability that the person was male is nothing but the row total corresponding to the category 'Male':

= 75.4% = 0.754

2. the probability that the person was a female aged 65 or older would be the cell corresponding to the row ' Female' and column '65 or older':

= 4.6% = 0.046

3. If we know the person killed was aged 65 or older, what is the probability they are female

= Pr( Female | 65 or older)

= Pr(Female and 65 or older) / P(65 or older)

= 0.046 / 0.17

= 0.271

iv. The probabilities used in I, ii and iii are, by definition - Marginal, Joint and Conditional probabilities respectively.

B) i. The figure in Exhibit 5 shows that out of young adults aged 17-25 who face fatalities, 77.2% i.e more than three fourth of them are Males. Hence, this result does support the conclusion outlined in the given comment.

ii. Suppose we are told that there are many more male drivers in the age range 17-25 than there are female drivers in this range.

Say, in the sample provided, there were 100 females and 100 males, with 22.8% of 100 = 23 female fatalities and 77.2% of 100= 77 male fatalities.

Now, as mentioned in the given scenario, suppose males were 150 in number (females being much lower, say 50), but the percentage of fatalities would still be the reported figures 22.8% of 50 = 11 in females and 77.2% of 150 = 116 (appx.) in males.

Still, we find that fatalities occur more in Males (116 / 150) than in females (11/50). Hence, this does not undermine the conclusion in the comment