Large Data Sets 4 and 4 A record the results of 500 tosses of a coin. Find the relative frequen

step 1 So we are considering four coins which are being tossed together. So now think if we toss four c

Large Data Sets 4 and 4 A record the results of 500 tosses...

Key Concepts

Law of Large Numbers

The law of large numbers is a foundational principle in probability that states as the number of trials increases, the observed relative frequencies of outcomes converge to their true theoretical probabilities. This concept explains why larger data sets are more reliable in evaluating the fairness of a coin or die.

Fairness in Probability Experiments

A coin, die, or any random device is considered fair or balanced if every outcome has an equal chance of occurring. In experiments involving a fair object, the relative frequencies of the outcomes tend to be approximately equal, corresponding to their theoretical probabilities.

Experimental Probability

Experimental probability is determined from observed outcomes in a series of trials. It reflects the likelihood of an event based on real experiment data rather than a theoretical model, and is instrumental in assessing how closely real-world outcomes align with expected probabilities.

Relative Frequency

Relative frequency is the proportion of times a specific outcome occurs compared to the total number of trials. It is calculated by dividing the count of a particular outcome by the total number of experiments, and it provides an empirical probability estimate based on actual data.

Transcript

00:01 So we are considering four coins which are being tossed together.

00:07 So now think if we toss four coins together, we will get either a head or a tail on each of them.

00:15 So what should be the number of total possible outcomes? it will be 2 into 2 into 2, 2 into 2, which is 16.

00:25 So let us see how that is going to happen.

00:28 So think in the first place if it comes ahead.

00:31 Then it can split into two options, right? either a head or a tail.

00:37 Like in the first place, if we are considered there is one head, then in the second one, we can have either a head or a tail.

00:46 Now, if we have a head, then again for the third coin, we can again have a head or a tail, or if the second coin was tail, then we can again have head or a tail for the third coin.

00:58 That means in the second coin, if it is head, then also we have third and third coin being head or tail same way even if we have the second coin being tail the third coin can still have head or tail and then for each of them it will be head tail head tail head tail and head tail well this thing is called oops a tree diagram okay now the catch here is we considered the first place will be head but the same thing will be there for second one t also so what will be the total number of sample space in the sample space will be h h h h h h h then h h then again see h h h t h, correct, so h, h, t.

02:07 Now, in the third place also, we will again have h t.

02:12 Now again, h, h, h, t, h t, h t, h t, h t, h t, t.

02:31 Same way now, just think we have to have the first place replaced with t, t, t, t, correct.

02:38 So we will have all these answers.

02:40 So t -h -h -h, then t -h -t -h -t -t -t -h -t -h -t -t -h -t -h -t -h and t -t -t -h -h and 4 times t.

03:05 Correct? now there is a random variable that assigns a number on each of these outcomes based on the number of heads.

03:19 So what is the number of heads here? so let's denote it with a red color.

03:25 So that it is easy...