Probability Normal Distribution Sampling and Sample size Benjamin Kamala kamala8086@gmail.com

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 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 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? 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 card is drawn at random from a deck of cards. Find the probability of getting the King of heart.  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. 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? 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?