Four fair coins are tossed simultaneously. Find the...

Key Concepts

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specified number. It is useful for calculating probabilities that involve phrases like 'at least' or 'not more than', by summing up the relevant individual probabilities.

Combinatorics

Combinatorics involves techniques for counting the number of ways events can occur. It is crucial for calculating probabilities in cases where the number of favorable outcomes (such as getting a specific number of heads) must be determined via combinations.

Binomial Distribution

The binomial distribution is applicable when dealing with a fixed number of independent trials, each with two possible outcomes (success or failure), and a constant probability of success. In scenarios like coin tossing, the number of heads obtained follows a binomial distribution, which simplifies the calculation of exact probabilities.

Probability Mass Function

The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to a specific value. It is fundamental in describing the distribution of outcomes for experiments with countable outcomes.

Random Variable

A random variable is a function that associates numerical values with outcomes of a random process. It provides a framework to quantitatively analyze results from an experiment, such as counting the number of heads in a series of coin tosses.

Transcript

00:01 Hello students, our question is four fair points are tossed simultaneously.

00:08 Find the probability function of the random variable x number of head and compute the probability of obtaining no heads precisely one hit at least one hit not more than c head.

00:26 So here four points are tossed, tossed simultaneously.

00:32 So we have to find the probability there no head appears one head appears are not more than three head appears so let's start our solution.

00:47 So here if p is the probability is the probability hat appears when a coin is tasked with the probability of p will be equal to 1 out of 2.

01:16 So appearing head is 1 out of 2.

01:18 Then q is the probability probability of tail appears.

01:25 The probability of q is also equal to 1 out of 2.

01:31 So here total 4 points are tasks simultaneously.

01:36 So n here will be equal to 4.

01:39 So we have n here is equal to 4 so we have to find the probability with no head appears the probability of no head appears means probability x is equal to 0 this will be equal to 4 c0 p that is p power p that is p is probability of success is one out of two one out of two power zero and probability of failure that will be equal to the tail appears to one out of two power 4 minus 0.

02:14 So this will be equal to 4 c0 will be equal to 1 .0, 1 out of 2 power 0 will also equal to 1.

02:22 So this will be equal to 1 out of 2 power 4.

02:26 So we apply the power 4 with 1 in 2 4 so we get 1 out of 16.

02:32 The probability date no hit appears is equal to 1 out of 16.

02:37 Now we have to find the probability that precisely 1 head.

02:42 So it means we have to find a probability of x is equal to 1.

02:46 So this will be equal to 4c1, 1 out of 2 power 1, and 1 out of 2 power 4 minus 1.

02:56 So this will be equal to 4c1 is equal to 4.

02:59 1 out of 2 power 1 is equal to 1 out of 2, and 1 out of 2 power 4 minus 1 will be equal to 1 out of 2 power 3.

03:09 So this will be equal to two one time, two two times...