Empirical Probability

Definition of Empirical Probability:

The experimental probability of occurring of an event is the ratio of the number of trials in which the event occurred to the total number of trials.

The empirical probability of the occurrence of an event E is defined as:

                          Number of trials in which event occurred
            P(E) =                       Total number of trials                  

Now we will solve the examples on different types of experiments and their outcomes such as tossing a coin, throwing of a die etc.,

Problems on empirical probability:

1. Let us take the experiment of tossing a coin.

When we toss a coin then we know that the results are either a head or a tail.
Thus, in tossing a coin, all possible outcomes are ‘Head’ and ‘Tail’.

Suppose, we toss a coin 150 times and we get head, say, 102 times.

Here we will find the probability of getting:

(i) a head and,

(ii) a tail

(i) Probability of getting a head:

Let E1 be the event of getting a head.

Then, P(getting a head)

           Number of times getting heads
= P(E1) =      Total number of trials        

= 102/150

= 0.68

(ii) Probability of getting a tail:

Total number of times a coin is tossed = 150

Number of times we get head = 102

Therefore, number of times we get tail = 150 – 102 = 48

Now, let E2 be the event of getting a tail.

Then, P(getting a tail)

           Number of times getting tails
= P(E2) =      Total number of trials        

= 48/150

= 0.32

Note: Remember, when a coin is tossed, then E1 and E2 are the only possible outcomes, and P(E1) + P(E2) = (0.68 + 0.32) = 1

2. Consider an experiment of rolling a die.

When we roll a die then the upper face of the die are marked as 1, 2, 3, 4, 5 or 6. These are the only six possible outcomes.
Suppose we throw a die 180 times and suppose we get 5 for 72 times.
Let E = event of getting 5 (dots).
Then, clearly, P(E) = 72/180= 0.40

3. Let us take the case of tossing two coins simultaneously.

When we toss two coins simultaneously then the possible of outcomes are: (two heads) or (one head and one tail) or (two tails) i.e., in short (H,H) or (H,T) or (T,T) respectively.

Let us toss two coins randomly for 100 times.

Suppose the outcomes are:

Two heads: 35 times

One head: 30 times

0 head: 35 times

Let E1 be the event of getting 2 heads.

Then, P(E1) = 35/100 = 0.35

Let E2 be the event of getting 1 head.

Then, P(E2) = 30/100 =0.30

Let E3 be the event of getting 0 head.

Then, P(E3) = 35/100 = 0.35.

Note: Remember, when two coins are tossed randomly, then E1, E2 and E3 are the only possible outcomes, and P(E1) + P(E2) + P(E3)

= (0.35 + 0.30 + 0.35)

= 1

9 Grade Math

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