A bag contains 9 red marbles, 8 white marbles, and 6 blue marbles. Randomly choose two marbles,

A bag contains 9 red marbles, 8 white marbles, and 6 blue...

Key Concepts

Probability

Probability is a branch of mathematics that measures the likelihood of events occurring. It provides a framework to quantify uncertainty and evaluate how likely an event is to happen based on a model of possible outcomes.

Sampling Without Replacement

Sampling without replacement involves selecting items from a set in such a way that once an item is chosen, it is not returned to the pool for subsequent selections. This impacts the probabilities of later selections as the total number of items decreases with each draw.

Conditional Probability

Conditional probability deals with the likelihood of an event occurring given that another event has already occurred. It is essential in scenarios where the outcome of a previous event influences the probability of subsequent events.

Multiplication Rule for Compound Events

The multiplication rule for compound events states that the probability of a sequence of events occurring together is the product of the probabilities of the individual events, adjusted by conditional probability when the events are dependent. This rule is key when analyzing sequential events in probability.

Law of Total Probability

The law of total probability is used to determine the overall probability of an event by considering all the distinct, mutually exclusive ways in which the event can occur and summing their individual probabilities. This is especially useful when the event of interest can occur under different conditions or through different pathways.

Transcript

00:01 In this video we are going to focus on the three sub -problems and here we can say that in the first part we are having that there are nine red marbles and we can say eight white marbles.

00:13 So it can be here written like this and then we have six blue marbles.

00:19 Therefore the total number of marbles will be here equals to we can write total and that is here 9 plus 8 plus 6 and that will be here 20.

00:30 So two marbles are selected randomly, one at a time and without replacement.

00:36 So as there are nine red marbles in 23 total marbles, so probability of the first marble chosen being red can be here written to be equal to it will be here 9 divided with 23.

00:51 And hence we can say the first chosen red marble is not replaced back into the bag.

00:58 So there are 8 white marbles out of the remaining 22 marbles.

01:02 And hence probability that the second marble is white can be here written to be equals to, we can say it will be 8 divided with 22.

01:13 Now the probability that the first marble is red and the second marble is white can be calculated as probability of, we can say first as red and second as white.

01:27 So it will be here written like this and since these two are independent events, so they will be here equals to 9 divided with 23, that is individual probability, right? so it will be here basically product of individual probabilities and that will be here coming around as 0 .1422.

01:48 And that is the answer for the first part of this very question.

01:53 Now let's see how we can find out the answer for the second part of this very question, right? so in the second sub -problem, we have the information that, let us write, that the two marbles to be of the same color, both can be red, white or blue.

02:11 Now there are nine red marbles in a total of 23 marbles, right? and the marbles are chosen without replacement.

02:18 So the probability that second marble is a red is equals to here that first one is red.

02:26 So here we can write it like this multiplied with it will be here probability that second one is red.

02:33 So it is here going to be equals to we can say that it will be here basically 9 divided with 23 and then it will be here 8 divided with 22.

02:44 So that is here equals to it will be coming around.

02:48 We can say that it will be 7 .2 divided with 506.

02:55 Now there are 8 white marbles in a total of 23 marbles, right? so the probability of second white marble can be here written as second and then here it will be white.

03:08 And that is here equals to probability that first one is white.

03:12 Multiplied with probability that second one is white.

03:16 So it will be here like this.

03:19 And hence we can say that it will be here equals to 8 divided with 23 multiplied with 7 divided with 22.

03:29 So it is here equals to 56 divided with 506.

03:36 And hence finally there are six blue marbles in a total of 23 marbles, right? so here we can say that probability of second blue marble can be written as it will be here probability of first blue and then it will be here multiplied with probability of second blue...

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