subject: Introduction To Independent Dependent Events [print this page] Introduction to independent dependent events:
Let we will 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 affects 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
Probability of Dependent Events
An event is a one or more possible outcomes from an experiment. An event is called dependent event if one event does affect the other event. An event consisting of more then one events is called compound event.
Probability of two dependent events
P(A and B) = P(A) P(A | B)
Probability of three dependent events
P(A and B and C) = P(A) (A | B) P(A | B | C)
Probability of Dependent Events - Examples
Example 1: Two fruits are drawn successively without replacement from a bag which contains 5 apples and 4 oranges. Find the probability that
a) The first fruit is drawn is apple and the second is orange
b) Both fruits are orange.
Solution:
a) Here the second event is dependent on the first.
P(A) = P(Apple) = 5/9
There are 8 fruits left and out of those 8, 4 of them are oranges.
So the probability that the second one is orange is given by:P(O | A) = P(Orange) = 4/8 = 1/2Dependent events, soP(A and O) = P(A) P(O | A) = 5/9 1/2 = 5/18.b) Dependent events.Therefore P(OO) = 4/9 3/8 = 12/72 = 1/6Example 2: A box contains 6 white balls, 4 black balls and 3 green balls. In each draw, a ball is drawn from the box and not replaced. In three draws, find the probability of obtaining white ball, black ball and green ball in that order.Solution:Dependent eventsP(W) P(B | W) P(G | B | W)= 6/13 4/12 . 3/ 11= 6/13 1/3 3/11= 18/429 = 6/143Example 3: A box contains 7 red marbles, 7 black marbles and 7 white marbles. Find the probability to choosing red marble and then, without replacing the red marble, choosing a white.Solution:These events are dependent events.The probability of choosing red marble is 7/21 = 1/3.The probability of choosing a white marble without replacement is 7/20.The probability of both is 1/3 7/20 = 7/60Probability of Dependent Events PracticeProblem 1: Find the probability of choosing 7 of diamond, and then, without replacing the 7 of diamond choosing a 7 in a deck of card.Answer: 884Problem 2: Find the probability of choosing king, and then, without replacing the king choosing a queen in a deck of card.Answer: 4/663= 1 / 6by: johnharmer
