Probability Rule of Independence, Statistical Note 4

Two events are said to be independent if the occurrence of one event does not affect the occurrence of the another. In that case the probability of the second event given the first event is equal to the probability of the first event.

I take an example from my statistical note 3 to show whether the sex of participants is independent of the food habit of the participants. Table 1 presents data on the number of training participants by sex of participants and food habit.

Table 1: Food habit of training participants variable by sex

Here, I will take the case of the joint probability of the first cell, P(VÇW). The sex of participant is independent of the food habit if P(V/W) is equal to P(V) or if P(W/V) is equal to P(W). The P(V/W) is equal to 12 divided by 16, 0.75. The P(V) is equal to 18 divided by 40, 0.45. P(V/W) is not equal to P(V). In another case, P(W/V) is 12 divided by 18, 0.67. The PW) is equal to 16 divided by 40, 0.4. Here also, P(W/V) is not equal to P(W). These prove that the food habit of the participant is dependent on the sex of the participant.

Now, I will manipulate the cell values to show that the food habit is independent on the sex of the participants. The number of women and men participants by food habits were made equal. It shows that the number of vegetarians or non-vegetarians whether women or men are equal meaning food habit is not changed irrespective of sex of the participants. P(V/W) needs to be equal to P(V) or P(W/V) needs to be equal to P(W) to be the food habit independent of sex of the participant.

Table 2: Food habit of training participants invariable by sex

P(V/W) is 8 divided by 16, that is 0.50 and P(V) is 20 divided by 40, which is equal to 0.50. It shows that P(V/W) is equal to P(V). Likewise, P(W/V) is equal to 8 divided by 20, equal to 0.40 and P(W) is equal to 16 divided by 40, which is also equal to 0.40. This also shows that P(W/V) is equal to P(W). These shows that the probability using the multiplication rule of probability without replacement, denoted by  P(VÇW) equal to P(V/W) multiplied by P(W) is equal to the probability using the multiplication rule of probability with replacement, denoted by P(VÇW) equal to P(V) multiplied by P(W). Thus, food habit is independent of the sex of the participant.