##### Question

In: Statistics and Probability

# A die has been "loaded" so that the probability of rolling any even number is 5...

A die has been "loaded" so that the probability of rolling any even number is

 5 27

and the probability of rolling any odd number is

 4 27

. (Assume the die is six-sided with each side numbered one through six.)

(a)

Find the following probabilities. Enter your probabilities as fractions.

 Pr(2 ∩ even) = Pr(even) = Pr(2 | even) =

What is the probability of rolling a 2, given that an even number is rolled?

(b)

Find the following probabilities. Enter your probabilities as fractions.

 Pr(3 ∩ (3 or 6)) = Pr(3 or 6) = Pr(3 | (3 or 6)) =

What is the probability of rolling a 3, given that a number divisible by 3 is rolled?

A bag contains 5 red balls and 7 white balls. Two balls are drawn without replacement. (Enter your probabilities as fractions.)

(a) What is the probability that the second ball is white, given that the first ball is red?

(b) What is the probability that the second ball is red, given that the first ball is white?

(c) Answer part (a) if the first ball is replaced before the second is drawn.

A red ball and 5 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting each of the following? (Enter your probabilities as fractions.)

(a) a red ball on the first draw and a white ball on the second

(b) 2 white balls

(c) 2 red balls

## Solutions

##### Expert Solution

Dice problem

Even numbers are 2,4,6 and odd numbers are 1,3,5

P(even) = 3/6 = 0.5 or 50%

P(odd) =3/6 =0.5

a) P(2 and even) = 1/3 = 0.33

P( even) =3/6 =1/2 =0.5

P(getting a 2 given that it is rolled to be an even number)

= P( 2 and an even number) / P (even number)

= (1/6)/(3/6) = 1/3 or 0.33%

b) P( a number 3) = 1/6

P( a multiple of 3) = 2/6 = 1/3

P(a number 3 is rolled given that it is a multiple of 3)

= P ( getting a 3 and a multiple of 3)/P(multiple of 3)

= (1/6) / ( 2/6) = 1/2 = 0.5

Bag full of balls

a) P( second ball is white given that first ball was red)

= P( first ball is red and second ball is white) / P(first ball is red)

= ( 5/12)×(7/11) / (5/12)

= 7/11

b) P ( 2nd ball is red given that 1st was white ball)

= P(1st ball is white and 2nd ball is red) / P(1st ball is white)

= (7/12)×(5/11) / (7/12)

= 5/11

c) if 1st ball is replaced then

P ( 2nd ball is white given that 1st one was red)

= (5/12)×(7/12) / (5/12) = 7/12

Red ball and white balls problem

a) P ( 1st ball is red and second ball is white)

= (1/6)×(5/5) = 1/6

b) P( 2 white balls) = (5/6)×(4/5) = 2/3

c) P ( 2 red balls) = (1/6)×(0/5) = 0

Because there exists only one red ball.

## Related Solutions

##### Suppose you have a three-sided die, the die is loaded so that the probability of 1...
Suppose you have a three-sided die, the die is loaded so that the probability of 1 or 2 coming out is the same and equal to one 4 while the third side has a probability of 1 two . If it is launched twice the die is X the random variable that returns the sum of the obtained results, write the table for the distribution function of probability p for the random variable X.
##### a) What is the probability of rolling a 5 with a balanced die? b) Find the...
a) What is the probability of rolling a 5 with a balanced die? b) Find the number of ways in which a person can select 3 stocks from a list of 5 stocks? c) What is probability of drawing a 5 from a well shuffled deck of 52 playing cards? d) In how many ways can a principal choose 3 of 45 members to review a student grade appeal?
##### Two dice are rolled. Determine the probability of the following. Rolling an even number or a...
Two dice are rolled. Determine the probability of the following. Rolling an even number or a number greater than 10
##### A die is weighted so that rolling a 1 is two times as likely as rolling...
A die is weighted so that rolling a 1 is two times as likely as rolling a 2, a 2 is two times likely as rolling a 3, a 3 is two times as likely as rolling a 4, a 4 is two times a likely as rolling a 5, and a 5 is two times as likely as rolling a 6. What is the probability of rolling an even number?
##### A question on my practice test is asking the probability of rolling an even number divisible...
A question on my practice test is asking the probability of rolling an even number divisible by 1.5 on a die. The answer it gives me is .333, which suggests that there are two numbers. However; between 1 and 6 only 3 and 6 are divisible by 1.5, and only one of those is even. Hence I get 1/6 =.1667. Am I missing something?
##### A loaded dice with the probability of rolling a one is 31 percent. You roll the...
A loaded dice with the probability of rolling a one is 31 percent. You roll the die a total of 7 times. (Answer using a decimal correct to three places.) a)Find the probability of rolling no ones. b)Find the probability of rolling exactly 3 ones. c)Find the probability of rolling 3 or 4 ones.
##### What is the probability of rolling 6, 6 sided die and achieving a score of 18?...
What is the probability of rolling 6, 6 sided die and achieving a score of 18? (6 die, each one is 6 sided)
##### You are interested in the probability of the outcome when rolling a six-sided die. Answer the...
You are interested in the probability of the outcome when rolling a six-sided die. Answer the following questions using p to denote the probability of this outcome. (a) You suspect that for this die the outcome does not occur with the same probability as with a fair die. Formulate the null and alternative hypotheses based on which you may make a statistical inference regarding your suspicion. (b) You suspect that for this die the outcome occurs with greater probability than...
##### A single die is rolled. Determine the probability of (show work) a) rolling a 6 given...
A single die is rolled. Determine the probability of (show work) a) rolling a 6 given the number is even b) rolling an even given the number is a 6 c) rolling a 5 given that the number is odd d) rolling a 2 or a 6 given that the number is even e) rolling a 4 given that the number is odd
##### For the following exercises, assume two die are rolled. What is the probability of rolling a pair?
For the following exercises, assume two die are rolled.What is the probability of rolling a pair?