The numbers 1, 2, 3, 4, and 5 are written on slips of paper, and 2 slips are drawn at random one

step 1 1, 2, 3, 4 and 5 are written on slips, and 2 slips are drawn at random. One at a time without re

The numbers 1, 2, 3, 4, and 5 are written on slips of paper,...

Key Concepts

Sample Space

The sample space is the complete set of all possible outcomes when an experiment is performed. In this type of problem, it represents all ordered pairs of numbers drawn from the slips of paper, which is fundamental for calculating probabilities.

Counting Principles

Counting principles, such as the multiplication rule, are used to determine the total number of outcomes in a multi-stage experiment. When drawing slips without replacement, the number of ways to obtain an ordered pair of numbers is calculated by multiplying the number of available options at each stage.

Probability

Probability is the measure of how likely an event is to occur, defined as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This concept is applied to compute the likelihood of specific events or combinations of events.

Ordered Outcomes

Considering ordered outcomes means that the sequence in which items are drawn matters. This is important in problems where the order affects the event definition, such as specifying the second number drawn.

Compound Events

Compound events involve combinations of two or more simple events connected by logical operators like 'and' or 'or'. Understanding how to handle compound events is crucial for calculating probabilities in scenarios where multiple conditions are satisfied.

Transcript

00:01 1, 2, 3, 4 and 5 are written on slips, and 2 slips are drawn at random.

00:07 One at a time without replacement, find the probabilities that both numbers are even.

00:13 So, firstly, probability that both even.

00:22 All right, so the sample space here is 20.

00:28 And why is the sample space 20? it is because we are taking without replacement.

00:38 So if we are going to do a square of a five, a five by five square would get 25, but we'll then eliminate five outcomes with uniform numbers.

00:54 That is one, one, two, two, three, three, four, four, five, because we are picking without replacement.

01:01 So that's why it becomes 20.

01:05 All right, enough of that explanation.

01:08 Let's come back to our question.

01:10 So our denominator will be 20 there.

01:13 So we want even numbers.

01:15 The even numbers that we have are 2 and 4 or 4 and 2.

01:19 So there are only two such cases out of 20.

01:23 And this will be reduced to 1 over 10.

01:27 The second question says, what is the probability that one of the numbers is even or greater than three.

01:35 So probability that the number is even or greater than three.

01:44 Now this is a typical ambiguous question because we are not, the question is not precise whether exactly one number or whether at least one number, number so you are forced to take any direction that you have interpreted the question to me so i'm going to take a direction that means exactly one number is even all greater than break so if i take that position this is the outcome that i'll have so what i need let me remove this once i know long i need them so what i need at this moment is a number which is even.

02:50 If i get an even number, then the other number should not be even.

02:57 Also, it should not be greater than three.

03:01 So that's the rule that i'm using.

03:03 So let me repeat the rule again.

03:06 I have to pick a number on a pair of numbers.

03:11 Only one number should either be even or greater than three.

03:19 Only one number.

03:21 If there are two of them that are even, it's out.

03:24 If there are two of them that are greater than three, then the combination is out.

03:31 So let's see what combinations are.

03:32 I'll start with one.

03:34 So say one and what? and two, it's fine because two is even.

03:41 Then the next number would be one.

03:45 One and three does not work because because one is less than three, and it's not even, and three is less than three.

03:58 So, so one and four would work, because four is even, then one and five would work, because five is greater than three, and then we can just reverse this order.

04:18 We can then say 2 and 1, 4 and 1 and 5 and 1.

04:26 All right, then that order is done.

04:28 We go to the other numbers.

04:30 We can start with 2.

04:35 2 and 1.

04:37 We already have it.

04:40 2 and 3.

04:42 It's okay.

04:46 2 and 4 does not work combination of 2 even numbers.

04:51 Then 2 and 5 does not work.

04:53 Work even number, and a number greater than five.

04:58 Two, okay, so the two are done.