Please help me solve this probability problems step by step with solution and re
ID: 3074438 • Letter: P
Question
Please help me solve this probability problems step by step with solution and reasoning. Please dont just write the solution. Thank you very much.
a)An experiment consists of asking five students at random if they wash their clothes with a particular brand of detergent. How many sample points can be formed if the answer from each student is ‘yes’ or ‘no’? Each sample point consists of five different answers.
b)How many sample points can be formed if three fair dice are tossed? Each sample point consists of three different numbers.
c)One die is tossed and the number of dots on the upper face of the die is observed. Find the probability to have ‘3’
dTwo dice are tossed and the numbers of dots on the upper faces of the dice are observed. Find the probability that the sum of the two numbers is equal to 7.
e)Two dice are tossed and the numbers of dots on the upper faces of the dice are observed. Find the probability that sum of the two numbers is equal to either 7 or 10.
f)Given that a roll of two fair dice has at least one three in a pair of numbers, what is the probability that the sum is four?
Explanation / Answer
a)
5 different students, answer is either "Yes" or "No". SO each student has 2 choices for response.
Number of Sample Points = 2 x 2 x 2 x 2 x 2 = 25 = 32
b)
3 fair dice are tossed. Each die has 6 different coutcomes.
Number of sample points = 6 (six options on 1st die) x 6(six options on 2nd die) x 6(six options on 3rd die) = 63 = 216
c)
A die is tossed. Probability that number of dots on upper face is '3'.
Only the face 3 of a die has '3' dots on it. So Probability that face 3 turns up = 1/6
d)
2 dice are tossed. Probability that the sum of two numbers is equal to 7.
Total number of possible outcomes = 6 x 6 = 36
Total number of outcomes in which the sum of two faces of dice is 7 :
{(3,4);(4,3);(1,6);(6,1);(2,5);(5,2)} = 6.
Probability that the sum of two numbers is equal to 7 = 6/36 = 1/6
e)
2 dice are tossed. Probability that sum of two numbers is equal to either 7 or 10: P("Sum is 7" or "Sum is 10") = P("Sum is 7") + P("Sum is 10")
The two events "Sum is 7" and "Sum is 10" are mutually exclusive.
P("Sum is 7") = 1/6 from previous part.
P("Sum is 10") = # {(5,5);(5,5);(4,6);(6,4)} = 4/36 = 1/9
P("Sum is 7" or "Sum is 10") = 1/6 + 1/9 = 5/18
g)
Event X: Roll of two fair dice has atleast one '3' in a pair of numbers.
Event Y: Sum of two numbers is 4
We have to find: P(Y | X)
Using conditional probability formula:
P(Y | X) = P(YX) / P(X)
P(X) = P("atleast one 3 in a pair of numbers") = 1 - P("No 3 in a pair of numbers")
P("No 3") = 52 / 62 = 25/36
P("Atleast one 3") = 1 - 25/36 = 11/36 = P(X)
P(YX) = P("atleast one 3 and sum of two nos. is 4")
Corresponding outcomes are: {(3,1);(1,3)}
P("atleast one 3 and sum of two nos. is 4"): = 2/36 = 1/18
P(Y | X) = (1/18) / (11/36) = 2/11