PCC | Mahmud | Statistics| statmania.info |
\(P (A) = \frac{n(A)}{n(S)}\)
\[\lim_{n(S) \to \infty} \frac{n(A)}{n(S)}\]
Three axioms
Say, S is sample space and A is an event
There are 15 cricketers in BD preliminary team. We got to select 11. C or P?
What is the probability that in a leap year, there are 53 Fridays?
Out of the natural numbers 10 through 30, a number is chosen randomly; what is the probability that the number is
What is the probability that the product of three positive integers chosen from 1 through 100 is an even number?
First Coin | |||
---|---|---|---|
H | T | ||
Second Coin |
H | HH | HT |
T | TH | TT |
Tossing a coin twice is equivalent to tossing two coins at once
What is the probability that
First Two Flips | |||||
---|---|---|---|---|---|
HH | HT | TH | TT | ||
Third Flip | H | HHH | HHT | HTH | HTT |
T | THH | THT | TTH | TTT |
What is the probability that
Tossing Two Dice at Once |
First Die | ||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
Second Die |
1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 | |
3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 | |
4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 | |
5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 | |
6 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 |
What is the probability that
Each rank has 13 cards.
3 cards are drawn from a pack of 52 cards. What is the probability that they are all Kings?
There are 4 Kings. We’ve to draw 3 cards.
If a card is drawn from a deck of 52 cards with 4 aces, what is the probability that an ace will not show up?
Let, P(A) = Ace appears
Two cards are drawn with replacement; What is the probability that they are
A card is drawn from a pack of 52 cards. What is the probability that it is
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
There are some balls in a box as below
Color | # Balls |
---|---|
White | 3 |
Black | 6 |
Red | 7 |
Green | 5 |
Yellow | 4 |
Violet | 9 |
Blue | 8 |
If three balls are drawn at random, what is the probability there are all red or green?
There are 9 red and 7 white balls in a box. 6 balls are picked randomly. What is the probability that 3 balls are red and 3 balls are white?
Which one is the answer?
Bayes Theorem
\(P(B|A)=\frac{P(A \cap B)}{P(A)}\)
Probability that it rains today is 40%, that tomorrow is 50%, and that on both days is 30%. If it rains today, what is the probability that it would rain tomorrow?
In a college, there are 100 students, of whom 30 play football, 40 play cricket, and 20 play both. A student is selected randomly. If he plays cricket, what is the probability that he plays football?
\(P(F)=0.3\)
\(P(C)=0.4\)
\(P(F \cap C)=0.2\)
\(P(F|C)=?\)
\(S=\) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
If a number is picked randomly and known it an even number, What is the probability that it is more than 6?
In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. The city has a facial recognition software. If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time. If the camera scans a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.
If the bell rings, what is the probability that a terrorist is caught?
About 99 of the 100 terrorists will trigger the alarm—and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So, the probability that a person triggering the alarm actually is a terrorist, is only about 99 in 10,098, which is less than 1%
Formulae
Think Why are they so?
P(R) = 0.4 and P(M) = 0.38
\(S_1\)={1,3,4,7,9,20}
\(S_2\)={12, 13, 14, 15, 16, 17, 18}
If a number is randomly chosen from each set, what is the probability that a prime number comes from \(S_1\) and a multiple of 3 from \(S_2\)?
Say, P = Prime from \(S_1\)
M = Multiple of 3 from \(S_2\).
\(P(P) = \frac 3 7\) and \(P(M)=\frac 3 7\)
What do we need to find out?
Cup 01 contains 2 black, 3 red, and 1 pink ball. Cup 2 contains only 1 red ball. A cup is selected randomly. Next, a ball is randomly chosen from that randomly selected cup and placed into the other cup. A ball is then drawn randomly from that second cup. Find the probability that the last ball drawn is a pink one.
If a senility researcher discovered that in a population of healthy and diseased elderly people, 14% of the people had senile dementia, 63% had arterioplerotic cerebral degeneration, and 11% had both. What is the probability that a person not having arterioplerotic cerebral degeneration has senile dementia?
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?
\(P(F)+P(S)+P(T)-P(F\cap S)-P(S\cap T)-P(F\cap T)+P(F\cap S \cap T\)
A card is drawn from each of two well-shuffled pack of cards. Find the probability that at least one of them is an ace.
Let,
A = Ace from 1st pack
B = Ace from 2nd pack
\(P(A \cup B)=?\)
\(P(A) = \frac{^4C_1}{^{52}C_1}\)
\(P(B) = P(A)\)
\(P(A\cap B) = P(A) \times P(B)\)
\(P(A\cap B)= \frac 1 3, P(A \cup B) = \frac 5 6, and \space P(A) = \frac 1 2\)
Find \(P(B)\) and \(P(B^c)\)
\(P(A)= \frac 1 2, P(B) = \frac 1 5, \text{and} \space P(A|B) = \frac 3 8\)
Find \(P(A \cap B), P(B|A)\), and \(P(A \cup B)\)
\(P(B|A) = \frac {P(A \cap B)}{P(A)}\)
\(P(A|B) = \frac {P(A \cap B)}{P(B)}\)
\(P(A \cap B) = P(A|B) \times P(B)\)
\(\therefore P(B|A) = \frac{P(A|B) \times P(B)}{P(A)}\)