Probability Flashcards
Total number of outcomes when tossing ‘n’ coins
2ⁿ
Total number of outcomes when throwing ‘n’ dices
6ⁿ
Complementary events
Impossible event
Event which is impossible and probability is zero
Another name of sure event
also definition
Certain event
Event which is sure and probability is one
Another name of certain event
also definition
Sure event
Event which is sure and probability is one
Total cards
52
how many suit in deck of cards
4
How many cards in each suit
13
Explain all suits of deck
Symbol and colour of club
♣ Black
Symbol and colour of spade
♠ Black
Symbol and colour of Diamond
♦Red
Face Cards
Jack, Queen, King (4x3=12)
Number Cards
2, 3, 4, 5, 6, 7, 8, 9, 10 (except ace)
[4x9=36]
Honor Cards
Ace + Face cards = Honor Cards
Ace, Jack, Queen, King
[4x4=16]
Equally Likely Events
Two or more events are said equally likely if each one of them has an equal chance of occurrence
Mutually Exclusive Events
(Doesn’t happen simultaneously)
Two or more events are mutually exclusive if the occurrence of each event prevents the every other event
Condition of mutually exclusive events
A∩B = ϕ
P(A∩B) = 0
Exhaustive events
All the events are exhaustive events if their union is sample space
E₁U E₂U E₃ …. Eₙ = S
P(E₁U E₂U E₃ …. Eₙ) = 1
When events are exhaustive, it guarantees that at least one of the events must occur because they collectively cover all possible outcomes of the experiment.
* When a sample space is distributed down into some mutually exclusive events such that their union forms the sample space itself, then such events are called exhaustve events
Sample space to tossing 4 coins simultaneously
Odd against
Odd in favour
What to to do in Random Drawn (Two or more)
Also random means
Use Combination
Not one by one (simultaneously)
Where to use combination in probability
Random Drawn (Two or more)
In mutual exclusive P(A∩B∩C)
0
In mutual exclusive P(AUBUC)
P(A) + P(B) + P(C)
Dead Heat
Draw
If E₁, E₂, E₃,…….Eₙ are mutual exclusive events then
Probability that any one of them occurs = ?
P(one of them occur) = P(E₁) + P(E₂) + P(E₃) +…….+P(Eₙ)
If E₁, E₂, E₃,…….Eₙ are the events and P(one of them occur) = P(E₁) + P(E₂) + P(E₃) +…….+P(Eₙ)
What it tells about events
1.)The events are mutually exclusive: The probability of their intersection is 0, so there’s no overlap to account for.
2.)The events are exhaustive: At least one of them must occur.
What does it mean if events E₁, E₂, E₃,…….Eₙ
are such that one must happen and only one can happen at a time?
They are Mutually Exclusive and Exhaustive Events.
If odd against ratio of A = m:n
Prob(A) = ?
(Derive it)
P(A) = n / (m+n)
If odd in favour of A = m:n
Prob(A) = ?
(Derive it)
P(A) = m / (m+n)
Dead heat is impossible means
Events are mutually exclusive
“a dead heat is impossible” effectively enforces the ________________, but it’s not necessarily the same as being ___________________.
“a dead heat is impossible” effectively enforces the mutual exclusivity condition, but it’s not necessarily the same as being exhaustive unless all possible outcomes are explicitly listed.
at least one of the events must occur means
Events are exhaustive
Conditional Probability
Let A and B are two events with same random experiment. Then the probability of occurrence of event A when event B has already occurred is called the conditional probability.
P(A/B) = Probability of event A given that B has already occured
P(B/A) = Probability of event B given that A has already occurred
P(A/B) = ? (Formula)
P(B/A) = ? (Formula)
read the answer
If there is a question of compound experiment, create sample space because the question of compound experiment solved by sample space
Keywords for multiplication law of probability
- successively
- One by one without replacement
- Dependent event
Multiplication law of probability
If A and B are two events associated with same random experiment. The probability of simultaneously occurrence of two events A and B is equal to the probability of one of the two events multiplied by the conditional probability of other.
P(A∩B) = P(AB) = P(A) P(B/A) or P(B) (A/B)
without replacement and combination point
Use combination in without replacement (no need to make cases)
in without replacement there is no difference between one by one and simultaneously
Where can’t we use combination
in replacement
Range of P(AUB) in terms of P(A) and P(B)
max(P(A),P(B))≤P(A∪B)≤P(A)+P(B)
Maximum value of P(AUB) and why?
Maximum of P(AUB) = maxP(A∪B)=min(1,P(A)+P(B))
Reason:
If A and B are mutually exclusive i.e. P(A∩B) =0
then P(A∪B)=P(A)+P(B).
Minimum value of P(AUB) and why?
Minimum of P(AUB) =max(P(A),P(B))
Reason:
If A s subset of B (or vice versa), meaning P(A∩B) = P(A) (or P(B)).
In this case, the union is essentially the larger event because one set is contained within the other.
If P(A) = 1, regardless of P(B)
What is P(AUB)
A is a certain event, so P(AUB) = 1 regardless of P(B)
How does the probability of A∩B compare to P(A)?
P(A∩B)≤P(A)
By definition,A∩B is a subset of A, so the probability of A∩B cannot exceed the probability of A.
This principle is part of the monotonicity of probability, which states that the probability of a subset is always less than or equal to the probability of the entire set.
Keyword for independent events
with replacement
Independent events
Two events are associated with a random experiment are said to be independent if the occurrence of one event does not effect the probability of occurrence of other.
P(A∩B) = P(A).P(B)
If A and B are independent events, then P(A/B) and P(B/A) = ?
and why?
If A and B are independent events,
then P(A/B) = P(A) [B se koi affect nahi padh rha A ko]
and P(B/A) = P(B)
Note for independent events
What is the formula for
P(onlyApasses) if A and B are independent?
P(onlyApasses)=P(A)⋅(1−P(B)).
What is the formula for
P(onlyApasses) if A and B are dependent?
P(onlyApasses)=P(A)⋅P(B’∣A),
where
P(B′∣A)=1−P(B∣A).
What is
P(onlyApasses) if A and B are mutually exclusive?
P(onlyApasses)=P(A),
because P(A∩B)=0 (A and B cannot occur together).
