CPhill

P (Blue Clothing or a Pair of Pants )  =  ????

P( Blue Clothing)  = 18/26

P( Pair of Pants)   = (14/26)

P( Blue Pair of Pants)  = 10/26     

So

Probability (Blue Clothing or a Pair  of Pants)  = P(Blue Clothing) + P (Pair of Pants) - P(Blue Pair of Pants)  =

18/26 + 14/26  -  10/26  =

22/26    =

11/13

Prob of  getting a quarter on the first draw  =  (5/10)

Prob of getting a nickel on the second draw  = (2/10)

So......the overall probability  is   (5/10) (2/10  =  10 / 100  =    1 / 10

The first one is known as a two-sided limit.....we are trying to find the limit as x approaches 1 from both sides

The next two are-one sided limits.....for the second, we are trying to find the limit as x approaches 1 from the left side

Finally....the last one means that we want to find the limit as x approaches 1 from the right side

Dependent

Independent  -  one roll  doesn't affect the next one

Independent

Dependent

"With replacemant"  usually indicates "independent".....

"Without replacement" usually  indicates "dependent"

Prob  of any one number on one spin  = 1/6

So...prob  of the same number twice in a row =  (1/6) (1/6)  =  1/36

Number of total coins =  9

Prob.  of  dime  and then quarter  =

(3/9)  (2/8)  =

(1/3) (1/4)  =

1/12

5.)

What is the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two? Express your answer as a common fraction.

To make                            Possible products                  #  of Numbers

0                               2nd, 3rd digits 0..1st digit 1 - 9        9

1                                           11                                        1

2                                           21                                        3

3                                           31                                        3

4                                           14 , 22                                 6                               

5                                           15                                        3

6                                           16   23                                 9                               

7                                            71                                       3

8                                            81   42                                9

9                                            91   33                                6

Prob  =   52 / 900    =   13 / 225

Using    

A little tricky....but......we can imagine the sine graph shifted down  one unit from the normal "parent" function  and having an amplitude of 3

So...far...we have this

y =  3sin (Bx)- 1

Note that a  "starting point"  on the curve is  (0, -1)  and a period of 1.5  puts us at  (-1, 4pi)

So.....  a period must be    4pi / 1.5  =  (8/3) pi

So.... to  find the number of periods in 2pi ,  ("B"), we can solve this

(2pi / (8/3 pi)  = B               

(6/3 pi) / (8/3 pi)  =

(6/3)(8/3)  =

6/8  =   3/4  =  B =  there are 3/4 periods in 2pi

So...our  function is

y  = 3sin [ (3/4)x ] -  1

Here's the graph : https://www.desmos.com/calculator/olu4zazirp

[ Note....other solutions are also possible ....this just seemed to be the easiest ]

So....

4.)

A bowl contains only red marbles, blue marbles and green marbles. The probability of selecting a red marble from the bowl is 3/13. The probability of selecting a blue marble from the bowl is 2/5. There are fewer than 100 marbles in the bowl. What is the probability of selecting a green marble and then a red marble from the bowl on the first two selections, assuming marbles are not returned to the bowl after being pulled?

There must be 65 marbles in the bowl because  5  and 13  have the lowest common multple  of  65....

3/13  =  15/65      and    2/5  =  26/65

So  

15 are red,  26 are blue and  65 - 15 - 26  =  24 are green

So...the probability of selecting green and then red  =

(24/65) (15/64)   =   (24/64) (15/65)   =  ( 3/8) (3/13)  =     9 /  104

1.)  There are 3 other aces  and 12 more hearts that will win the small prize

And there are 52 possibilities when we draw a card from the deck.....

So....the probability  of winning a small prize  =

Favorable outcomes  /  Total possible outcomes   =

[ 12 + 3 ] / 52  =

15 / 52