Conditional Probability
If \(B\) occurs, what is the probability that A occurs?
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
Conditional Probability Example
\(P(A)\)
Multiplicative Laws
- If A & B are independent, \(P(A \cap B) = P(A) \times P(B)\)
- Otherwise \(\uparrow\)
- \(P(A \cap B) = P(B) \times P(A|B)\)
Check dependency
S = {1,2,3,4,5,6}
- Case -1. A = {1,3,5}, B = {2,4,6}
- Case -2. A = {1,3,5}, B={1,3,4,6}
Conditional - Two bags
A bag I contains 4 white and 6 black balls while another Bag II contains 4 white and 3 black balls. One ball is drawn at random from one of the bags, and it is found to be black. Find the probability that it was drawn from Bag I.
Expansion of P(A ∩ B̅)
\(P(A \cap \bar B) = 1 - P(A ∩ B)\)
WHY?
Expansion of A
- \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
- \(P(A) = P(A \cap B) + P(A \cap \bar B)\)
- Expand more
- \(P(A) = P(A|B) \times P(B) + P(A|\bar B) \times P(\bar B)\)
Conditional: Job Completion
A person has undertaken a job. The probability of completing the job on time if it rains is 0.44, and the probability of completing the job on time if it does not rain is 0.95. If the probability that it will rain is 0.45, then determine the probability that the job will be completed on time.
Conditional: 3 Urns
There are three urns containing 3 white and 2 black balls, 2 white and 3 black balls, and 1 black and 4 white balls, respectively. There is an equal probability of each urn being chosen. One ball is equal probability chosen at random. What is the probability that a white ball will be drawn?
Types of Selections
- At Once/Toegther
- One by one
- without Replacement
- With Replacement
Drawn At Once/Toegther
A box has 6 red, 5 white, and 4 green balls. 2 balls are drawn at random. What is the probability that –
- both are red
- \(P(RR)\)
- they are different colors
- \(P(RW \cup RG \cup WG)\)
- Does order matter? (\(WG = GW?)\)
Same-Different Colors
A pot contains 3 white, 4 red, and 5 blue balls. Three balls are drawn at random. Find the probability that the balls are
- different colors
- same colors
- \(i \rightarrow P(WRB)\)
- \(ii \rightarrow P(WW \cap RR \cap BB)\)
- What if they are drawn with replacement?
One by One
A box has 6 red, 5 white, and 4 green balls. 2 balls are drawn at random. What is the probability that –
With and Without Replacement
In a box, there are 5 blue marbles, 7 green marbles, and 8 yellow marbles. If two marbles are randomly selected, what is the probability that both will be green or yellow, if taken
with replacement
without replacement
To add or multiply?
Simple \(\cap | \cup?\)
Card Problem
2 cards are drawn from a pack of 52 cards without replacement.
- P(Kings of same color)
- P (No king)
Set Theory - Pass/Fail
In a class of 40 students, 23 passed in Mathematics, 25 passed in English, and 10 failed in both the subjects. A student is randomly selected.
- P(passed in at least one subject)
- P(passed in both)
- P(passed in Mathematics alone)
- \(i \rightarrow 1 - P(\bar M \cap \bar E\)
- \(ii \rightarrow P(M) + P(E) - P(M \cup E)\)
- \(iii \rightarrow P(M) - P(M\cap E)\)
Set Theory-Newspaper
40% people read the Jugantor and 25% read the Amar Desh, while 20% read both.
- P(At least one)
- P(If Amar Desh, then Jugantor)
Intersection vs Multiplication
A number is chosen between 2 and 440. What’s the probability that its a square number and multiple of 2
- Let \(A =\) Square number, \(B =\) multiple of 2
- \(P(A \cap B) \ne P(A) \times P(B)\)
Solving by complementary method
A candidate applied for three posts in an industry, having 3, 4, and 2 candidates respectively. What is the probability of getting a job by that candidate in at least one post?
- Let \(P(A) = 1/3, P(B) = 1/4, P(C) = 1/2\)
- \(P(A \cup B \cup C) = ?\)
- Long formula
- Or just \(1-P(A) \cdot P(B) \cdot P(C)\)
- Coz \(P(A \cup B \cup C) = 1 - P(A \cap B \cap C)\)
