In: Statistics and Probability

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

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.

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 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 number greater than 10

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 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 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? (6 die, each one is 6 sided)

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 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?

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