Lesson FURTHER - Probability2 - Examples
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<b>EXAMPLE</b> A bag holds 3 red and 5 blue marbles. Picking one at random and then replacing it into the bag, find the following probabilities: 1. P(picking a red) 2. P(picking a red and then another red) 3. P(picking a red and a blue) solutions: 1. P(RED) = 3/8 2. P(RED then RED) = P(RED) AND P(RED) P(RED then RED) = 3/8 * 3/8 P(RED then RED) = 9/64 3. P(RED and a BLUE). note...no order specified. P(RED and a BLUE) = [P(RED) AND P(BLUE)] + [P(BLUE) AND P(RED)] P(RED and a BLUE) = [3/8 * 5/8] + [5/8 * 3/8] P(RED and a BLUE) = [15/64] + [15/64] P(RED and a BLUE) = 30/64 P(RED and a BLUE) = 15/32 <b>EXAMPLE</b> A bag holds 3 red and 5 blue marbles. Picking one at random and NOT replacing it into the bag, find the following probabilities: 1. P(RED) 2. P(another RED) 3. P(BLUE) solutions: 1. P(RED) = 3/8 note: there are now 2 red marbles left in the bag. Total is 7 marbles. 2. P(another RED) = 2/7 note: there is now 1 red marble left in the bag. Total is 6 marbles. 3. P(BLUE) = 5/6 <b>Probability Trees</b> Sometimes there are 2 things going on that you want to keep track of. The best way to do this is do draw a Probability tree. <b>EXAMPLE</b> A man goes walking every Sunday. The probability of it raining on that day has been calculated to be 0.15. The man has a particular walking stick which he uses to walk with on 3 out of 4 times. Find the probability that 1. the man has his stick when it rains and 2. the man does not have his stick whilst it is dry. Now looking at these without some sort of visual help is dangerous, since you will make mistakes. And odds on, the mistake will come in your exam. So, get used to drawing a quick diagram, a probability tree. So, Let S = man has stick and xS = man does not have stick also, let R = it rains and xR = it does not rain. <b> apologies for the bad drawing. Hopefully you can take my picture and apply it to a proper Probability tree</b> _____________________0.15________ R ---> SR = 0.75 * 0.15 = 0.1125 ___0.75_______ S _____________________0.85________ xR ---> SxR = 0.75 * 0.85 = 0.6375 _____________________0.15________ R ---> xSR = 0.25 * 0.15 = 0.0375 ___0.25_______ xS _____________________0.85________ xR ---> xSxR = 0.25 * 0.85 = 0.2125 Note: Add up all the probabilities at the right hand side... these are all the possible outcomes, so they should add up to 1. If they do not you know you have gone wrong somewhere, so go back and check your working. Now, to answer the questions: 1. the man has his stick when it rains. This is SxR --> 0.6375 2. the man does not have his stick whilst it is dry. This is xSR --> 0.0375 <b>Summary</b> These 3 lessons on probability introduce the concepts in Probability and give examples on its various stages. The examples given are classic versions that should give you a good basic understanding of the topic.
