Definition n of product rule
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We shall begin by drawing up a table of the data set;
The Probability of selecting someone in the 22 - 29 age bracket is derived as
P[22 - 29] = No of req outcomes/No of possible outcomes
P[22 - 29] = 258/360
P[22 - 29] = 0.7166
The probability of selecting someone who refused to respond is derived as
P[Dont respond] = 42/360
P[Dont respond] = 0.1166
Note however that these are dependent events. That means the outcome of one event directly affects that of the other event. In this experiment, you are required to select one person from the entire sample size. Hence the probability of P[22 - 29] OR P[Dont respond] is derived by adding both probabilities and this results in;
The probability of getting someone who falls within the 22 - 29 age bracket or someone who refuses to repond is described as non mutually exclusive events. Mutually exclusive events are those that cannot possibly occur in one experiment. For example you cannot get a King of Hearts OR an Ace of Hearts in one draw from a standard deck of 52 cards. However when you can get either one or the other in one single experiment, then this is refered to as non mutualy exclusive events. So the probability of drawing from a standard deck of 52 cards, a King OR an Ace is an example of non mutually exclusive events because you can get either one or the other with just a single draw of a card. In this experiment, you need to select just one person and the probability of the person being either within the age of 22 - 29 OR being a non-respondent is a non mutually exclusive event. You can select someone randomly and he'll be within 22 - 29 years of age, and its more than likely that that person did not respond to your survey. Fact is,out of those who were 22 - 29, some responded, and some did not. So they all stand a chance of being selected, regardless of how slim their chances were.
Hence, the results should be
P[22 - 29 or Dont respond] = P[22 - 29] + P[Dont respond] - P[(22 - 29) * (Dont respond)]
P[22 - 29 or Dont respond] = ([0.7166] + [0.1166]) - [0.7166 * 0.1166]
P[22 - 29 or Dont respond] = (0.8332) - 0.08355
P[22 - 29 or dont respond] = 0.74965
The results show that the probability of choosing someone who is within the 22 - 29 age bracket or someone who did not respond is 0.74965 which as a percentage is 74.965% (approximately 75%)
