Principle of Mathematical Induction

Principle of Mathematical Induction:

Let us discuss about the independent and dependent events in probability. If the two events should be said to be dependent when occurrence or outcome of first event affects the occurrence or outcome the second event. Therefore, their probability will changed in independent and dependent events.

If two events should be called independent when the outcome of first event should not affect the outcome of second event.

Independent Dependent Events-dependent Events:

* Let us consider more than two events that are dependent.
* When p1 should be probability of first event, p2 be the probability that happens after first event and p3 will be the probability that occurs after first and second events.
* Then probability of all events will happen will be the product p1 - p2 - p3.


Example problem:

A bag has 6 blue balloons, 4 green balloons and 2 black balloons. In every draw, a balloon is drawn from the bag and not replaced. In three draws, find the probability of obtaining blue, green and black in that order.

Solution:

Given, Blue balloons = 6

Green balloons = 4

Black balloons = 2

Total number of balloons = 6 + 4 + 2 = 12

Here, the three events are dependent.

So the probability = ( 6 / 12 ) × ( 4 / 11 ) × ( 2 / 10 )

= ( 1 / 2 ) × ( 4 / 11 ) × ( 1 / 5 )

= 4 / 110

= 2 / 55
Independent Dependent Events-independent Events:

* Two events P and Q are called independent when reality that P occurs should not affect the probability of Q happening.


Example problem:

A die should be tossing two times. What will be the probability of getting 2 or 4 on first toss and 1, 3, or 5 in second toss.

Solution:

Let, probability of getting 2 or 4 is P(E1) and probability of getting 1,3 and 5 will be P(E2).

Now, P(E1) = P (2 or 4) = 2 / 6 = 1 / 3

P(E2) = P (1,3 or 5) = 3 / 6 = 1 / 2

Here, they are independent events.

Therefore, P(E1 and E2) = P(E1) × P(E2)

= 1 / 3 × 1 / 2

= 1 / 6

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