Example 18

EXAMPLE 18 Having more information often affects the probability of an event

Table 19 contains the Academy Award winners for Best Actress and Best Actor for 2009–2014, along with their ages at the time of the award.

Table 5.52: Table 19 Oscar-winning actresses and actors, and their ages

Year	Best actress	Film	Age	Best actor	Film	Age
2009	Kate Winslet	The Reader	33	Sean Penn	Milk	48
2010	Sandra Bullock	The Blind Side	45	Jeff Bridges	Crazy Heart	60
2011	Natalie Portman	Black Swan	29	Colin Firth	The King's Speech	50
2012	Meryl Streep	Iron Lady	62	Jean Dujardin	The Artist	39
2013	Jennifer Lawrence	Silver Linings Playbook	22	Daniel Day- Lewis	Lincoln	55

Table 5.52: Source: www.oscars.org.

Table 20 is a contingency table, summarizing the information in Table 19, providing the counts of the performers' genders and whether the performer was under age 40.

Table 5.53: Table 20 Contingency table of Oscar-winning performers

	Female	Male	Total
Under age 40	3	1	4
Age 40 or older	2	4	6
Total	5	5	10

Now, if we choose a performer at random from Table 20, the probability of choosing a female is . But what if we were given the extra information that the performer is age 40 or older? How does this extra information affect the probability of selecting a female?

Solution

Notice that when we are given that the person is age 40 or older, we may restrict our attention to the performers who are age 40 or older in Table 20 (highlighted). In other words, this extra information reduces the number of possible outcomes in the sample space from the 10 performers to the 6 performers who are age 40 or older. Of these six performers, two of them are female. Thus, the probability of selecting a female, given that the performer is age 40 or older, is . The extra information we were given changed the probability of selecting a female, from to .