Probability – An Experimental Approach

Experimental probability is the result of probability based on the actual experiments. It is also called the Empirical Probability.

In this probability, the results could be different, every time you do the same experiment. As the probability depends upon the number of trials and the number of times the required event happens.

If the total number of trials is ‘n’ then the probability of event D happening is

\mathrm{P(D)=\dfrac{Number\ of\ trials\ in\ which\ the\ event\ happened}{The\ total\ number\ of\ trials}}

Examples

1. If a coin is tossed 100 times out of which 49 times we get head and 51 times we get tail.

a. Find the probability of getting head.

b. Find the probability of getting tail.

c. Check whether the sum of the two probabilities is equal to 1 or not.

Solution

a. Let the probability of getting head is P(H)

\mathrm{P(H)=\dfrac{Number\ of\ trials\ in\ which\ the\ head\ comes}{The\ total\ number\ of\ trials}}=\dfrac{49}{100}

b. Let the probability of getting tail is P(T)

\mathrm{P(H)=\dfrac{Number\ of\ trials\ in\ which\ the\ tail\ comes}{The\ total\ number\ of\ trials}}=\dfrac{51}{100}

c. The sum of two probability is

\mathrm{= P(H) + P(T)=\dfrac{49}{100}+\dfrac{51}{100}=\dfrac{100}{100}=1}

Impossible Events

While doing a test if an event is not possible to occur then its probability will be zero. This is known as an Impossible Event.

Example

We cannot throw a dice with number seven on it.

Sure or Certain Event

While doing a test if there is surety of an event to happen then it is said to be the sure probability. Here the probability is one.

Example: 1

It is certain to draw a blue ball from a bag contain a blue ball only.

This shows that the probability of an event could be

0\le P(E)\le1

Example: 2

There are 5 bags of seeds. If we select fifty seeds at random from each of 5 bags of seeds and sow them for germination. After 20 days, some of the seeds were germinated from each collection and were recorded as follows:

Bag12345
No. of seeds germinated4048423941

What is the probability of germination of

(i) more than 40 seeds in a bag?

(ii) 49 seeds in a bag?

(iii) more than 35 seeds in a bag?

Solution:

(i) The number of bags in which more than 40 seeds germinated out of 50 seeds is 3.

P (germination of more than 40 seeds in a bag) =\dfrac{3}{5}=0.6

(ii) The number of bags in which 49 seeds germinated = 0.

P (germination of 49 seeds in a bag) =\dfrac{0}{5}=0 .

(iii) The number of bags in which more than 35 seeds germinated = 5.

So, the required probability =\dfrac{5}{5}=1 .

Elementary Event

If there is only one possible outcome of an event to happen then it is called an Elementary Event.

\text{P (H) + P (T) = 1}

\mathrm{P\ (H) + P(\bar H)=1} (where \mathrm{\bar H} is ‘not \text H’.  

\mathrm{P\ (H) - 1=P(\bar H)}

\mathrm{P\ (H)} and \mathrm{P(\bar H)} are the complementary events.

Example

What is the probability of not hitting a six in a cricket match, if a batsman hits a boundary six times out of 30 balls he played?

Solution

Let \text D be the event of hitting a boundary.

\mathrm{P(D)=\dfrac{Number\ of\ times\ the\ batsman\ hits\ the\ boundary}{Total\ number\ of\ balls\ he\ played}}

=\dfrac{6}{30}=\dfrac{1}{5}=0.2

So the probability of not hitting the boundary will be

 \mathrm{P(\bar D)=1-0.2=0.8}

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