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Probability
Normal Distribution
Sampling and Sample size
Benjamin Kamala
[email protected]
Probability
Proportional of times the event
occurs in a long series of random
numbers
• E.g. tossing a coin once
• P(H) = ½,
• P(T) = 1/2
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Tossing a dice (1,2,3,4,5,6)
P(1)=1/6, p(2)=1/6, p(3)=1/6……
Even numbers, E: 2,4,6
Prime number, O: 2,3,5
P(E)=3/6=1/2
P(O) = 1/2
Note
• Take values from 0 to 1
• P= 0=event never occur, P=1=event certainly
occurs
• The sum of all probabilities = 1
Calculation
• Additional rule:(A or B)=P(A)+P(B)-P(A and B)
Independent events
• Occurrence of one event does not affect in
anyway the occurrence of the other
• When a coin and Dice are tossed together the
occurrence of any number on the dice does
not affect the occurrence of H/T
• Rule: P(A and B) = P(A)*P(B)
If the coin and dice are tossed
together:
• Find the probability of getting a H and even
number (E)
• P(H)=1/2
• P(E)=3/6=1/2
• P(H and E) = ½*1/2=1/4
• A pack of 52 playing cards: 13 spades, 13
hearths, 13 flowers, 13 Diamonds
• If a two cards are picked (one at a time) in
random manner what is the probability that
both are Diamonds?
• If a two cards are picked (one at a time) in
random manner what is the probability that
one is a Diamonds and another is a Spade?
Example
• If Joshua and Hassan attend Biostatistics
lectures only once in a month and there is one
Biostatistics lecture per week; what is the
probability of finding both of them in a given
lecture
• What is the probability of finding either of
them in a given lecture
• Mutually exclusive events: two events never
occur together
• P(A and B) = 0
• E.g. when the coin is tossed once P(HnT) = 0
• Two coins are tossed; find the probability that two heads are obtained.
• Question 3: Which of these numbers cannot be a probability?
a) -0.00001
b) 0.5
c) 1.001
d) 0
e) 1
20%
• Two dice are rolled, find the probability that the sum is
a) equal to 1
b) equal to 4
c) less than 13
Examples
• A die is rolled and a coin is tossed, find the
probability that the die shows an odd number
and the coin shows a head.
• A card is drawn at random from a deck of
cards. Find the probability of getting the 3 of
diamond
• A jar contains 3 red marbles, 7 green marbles
and 10 white marbles. If a marble is drawn
from the jar at random, what is the probability
that this marble is white
• Question 1: A die is rolled, find the probability that an
even number is obtained
• Question 2: Two coins are tossed, find the probability
that two heads are obtained.
• Question 4: Two dice are rolled, find the probability that
the sum is
a) equal to 1
b) equal to 4
c) less than 13
• Question 5: A die is rolled and a coin is tossed, find the
probability that the die shows an odd number and the
coin shows a head
• Question 6: A card is drawn at random from a deck of
cards. Find the probability of getting the 3 of diamond.
• Question 7: A card is drawn at random from a deck of
cards. Find the probability of getting a queen.
• Question 8: A jar contains 3 red marbles, 7 green marbles and 10 white
marbles. If a marble is drawn from the jar at random, what is the
probability that this marble is white?
• Question 9: The blood groups of 200
people is distributed as follows: 50 have
type Ablood, 65 have B blood type, 70
have O blood type and 15 have
type AB blood. If a person from this group
is selected at random, what is the
probability that this person has O blood
type?
• Tickets numbered 1 to 20 are mixed up
and then a ticket is drawn at random.
What is the probability that the ticket
drawn has a number which is a multiple of
3 or 5?
• A bag contains 2 red, 3 green and 2 blue
balls. Two balls are drawn at random.
What is the probability that none of the
balls drawn is blue?
• Tickets numbered 1 to 20 are mixed up and then a ticket is
drawn at random. What is the probability that the ticket
drawn has a number which is a multiple of 3 or 5?
• A bag contains 2 red, 3 green and 2 blue balls. Two balls are
drawn at random. What is the probability that none of the
balls drawn is blue?
• What is the probability of getting a sum 9 from two throws of
a dice?
• Three unbiased coins are tossed. What is the probability of
getting at most two heads?
• A number X is chosen at random from the numbers -3, -2, -1,
0, 1, 2, 3. What is the probability that |X|<2
• Two brother X and Y appeared for an exam. Let A be the event
that X is selected and B is the event that Y is selected.
– The probability of A is 17 and that of B is 29. Find the probability that
both of them are selected.
Home Work
a) A card is drawn at random from a deck of cards. Find
the probability of getting the King of heart.
b) A bag contains 21 toys numbered 1 to 21. A toy is drawn and
then another toy is drawn without replacement. Find the
probability that both toys will show even numbers.
c) A speaks truth in 75% of cases and B in 80% of cases. In what
percent of cases are they likely to contradict each other in
narrating the same event?
d) What is the probability that a number selected from
numbers 1,2,3,...,30, is prime number, when each of the
given numbers is equally likely to be selected?