Consider the sample space containing all of the possible...

Consider the sample space containing all of the possible outcomes of rolling two dice. Let S represent the event that the sum of the two faces showing is exactly 7. Let T represent the event that the left die shows a 2. Thoroughly describe how you would find the answer to the following questions including all formulas and calculations, explaining what you're doing as you go. 1. What is P(S)? 2. What is P(S|T)? 3. Are S and T mutually exclusive? Why or why not? 4. Are S and T independent? Why or why not? 5. Given that the left die shows a 4, 5, or 6, what is the probability that the sum is 5?

Transcript

00:01 It is given that in rolling the two dices, the total number of possible outcomes would be 36.

00:10 And it is given that the event s would be the sum of number 7 in the rolling of two dice.

00:19 For s, the possible evens are 1 .6.

00:24 In this case, adding 1 and 6 gives the sum of 7.

00:27 Similarly, the other possible outcomes are 2 .5, 3 .4, 4 .3, 5, 5 comma 2 and 6 .1.

00:40 This implies the number of outcomes for the event s would be 6.

00:46 And it is given that event 2 would be the number 2 will be showing in the left die.

00:54 And this implies for t the number of possible outcomes are 2 .1 2 .2 2 .3 2 .3 2 .4 2 .5 and 2 .6.

01:10 In this case all the left dice are showing number 2.

01:15 And this implies the number of outcomes for the event t is also 6.

01:20 Now the part 1 of the probability of the probability of the even.

01:24 Event s is calculated as the number of outcomes for the event s divided by the total number of outcomes.

01:33 The number of outcomes for event s is calculated as 6 divided by the total outcome of 36.

01:39 Now simplifying this gives the value of 0 .1 -667.

01:45 Now in question 2, we have to find the probability of the event s given t.

01:51 Using the conditional probability, this is calculated as.

01:54 As number of outcomes of the event s intersection t divided by the number of outcomes of the event t.

02:03 Now the event s intersection t implies the common outcome in both the event of snt.

02:13 The only common event which is in both the event yes and the event t is the 2 comma 5...

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