We can calculate the probability of an event from a sample space easily by using the probability formula for equally likely events. What if there are events in the sample space that are totally unrelated? What if there are two coins? The result of one experiment is totally independent of the result of another experiment. Such is the case of the independent events. In the following section, we will see how we can find the probability of these events. Let us see!

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## Independent Events

In the language of mathematics, we can say that all those events whose probability doesn’t depend on the occurrence or non-occurrence of another event are Independent events. For example, say we have two coins instead of one. If we flip these two coins together, then each one of them can either turn up a head or a tail and the probability of one coin turning either a head or a tail is totally independent of the probability of the other coin turning up a head or a tail. Such events are known as independent events.

The probability of an independent event in the future is not dependent on its past. For example, if you toss a coin three times and the head comes up all the three times, then what is the probability of getting a tail on the fourth try? The answer is simply 1/2. Whatever

Example 1: Out of the following examples, which represents an independent event?

A) The probability of drawing an Ace from a well-shuffledÂ pack of 52 cards, twice.

B) Probability of drawing a King from a pack of 52 cards and an Ace from another well-shuffled pack of 52 cards.

C) Two queens which we draw out of a well-shuffled pack of 52 cards.

D) All of the above events are examples of independent events.

Answer: Let us see each of the options one by one. In option A), the two of events are drawing an ace and then drawing another ace. When we draw the first ace, we have one event in our favour and 52 in total. So the probability is 1/52. For the second draw, there is 1 less card in the deck, so these events in which we have only one pack of cards can’t be independent events.

Hence only the option B)Â Probability of drawing a King from a pack of 52 cards and an Ace from another well-shuffled pack of 52 cards, is correct.

## Rule Of The Product

The total probability of events that are independent is found out by multiplying the probability of these events. Let us see with the help of examples:

Example 2: Two coins are flipped simultaneously. What is the probability of getting heads on either of these coins?

Answer: First thing that you realise is that these are independent events. Once you do that, move on to find the probability of each individual event. Let us call the first coin toss as E and the second coin toss as F. Therefore we can write: P (E) = 1/2 i.e. probability of getting a head on the first coin toss = 1/2.

Similarly, the probability of getting a head on the second coin’s toss = 1/2. In other words, we can write that P (F) = 1/2.

Now we have to calculate the probability of both these events happening together. hence we use the rule of the product. If P is the probability of some event and Q is the probability of another event, then the probability of both P and Q happening together is PÃ—Q.

Hence the probability that either of the two coins will turn up a headÂ = 1/2Â Ã— 1/2 = 1/4

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Example 3: A die is cast twice and a coin is tossed twice. What is the probability that the die will turn a 6 each time and the coin will turn a tail every time?

Answer: Each time the die is cast, it is an independent event. The probability of a getting a 6 is = 1/6. So the probability of getting a 6 when the die is cast twice = 1/6 Ã— 1/6 = 1/36

Similarly the probability of getting a tail in two flips that follow each other (are independent) = (1/2)Ã—(1/2) = 1/4

Therefore as the two events i.e. casting the die and tossing the coin are independent, and the probability of both the events = (1/36)Ã—(1/4) = 1/144.

The Rule of products is only applicable to the events that are independent of each other. The product gives the total probability of such events. In other words, the probability of all such events occurring is what we get from the product of probabilities.

On the other hand, the sum of probabilities gives the total probability of mutually exclusive events. If one event P occurs and thus prevents the occurrence of another event Q, then the probability of both these events occurring is = Probability of P + Probability of Q.

## Practice Problems:

Q 1: A die is cast 6 times. What is the probability that each throw will return a prime number?

A) 1/ 32Â Â Â Â Â Â Â B) 1/1296Â Â Â Â Â Â C) 1/64Â Â Â Â Â Â Â D) 1/1666

Ans: C) 1/64

Q 2: A coin is flipped six times. What is the probability of getting a head each time?

A)Â 1/64Â Â Â Â Â Â Â B) 1/1296Â Â Â Â Â Â C)Â 1/ 32 Â Â Â Â D) 1/1666

Ans: A) 1/64

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