A batch of 25 injection-molded parts contains 5 that have suffered excessive shrinkage. Round your answers to four decimal places (e. g. 98.7654). (a) If two parts are selected at random, and without replacement, what is the probability that the second part is one with excessive shrinkage? (b) If three parts are selected at random, and without replacement, what is the probability that the third part is one with excessive shrinkage?

(a) The probability that the second part is the one with excessive shrinkage is 0.2000.

(b) The probability that the third part is the one with excessive shrinkage is 0.2000.

Step-by-step explanation:

Let the variable X ₙ denote the nth part that has  suffered excessive shrinkage.

(a)

It is provided that two parts are selected at random.

The parts are selected without replacement.

Now, for the two parts selected it is possible that either both have excessive shrinkage or only the second part has excessive shrinkage.

The probability that the second part is the one with excessive shrinkage is:

P (X₂) = P (X₂ ∩ X₁) + P (X₂ ∩ X₁')

                        (without replacement)

Thus, the probability that the second part is the one with excessive shrinkage is 0.2000.

(b)

It is provided that three parts are selected at random.

The parts are selected without replacement.

Now, for the three parts selected it is possible that either all three have excessive shrinkage or any two of the three has excessive shrinkage or only the third part has excessive shrinkage.

The probability that the third part is the one with excessive shrinkage is:

P (X₃) = P (X₃ ∩ X₂ ∩ X₁) + P (X₃ ∩ X₂ ∩ X₁')

                 + P (X₃ ∩ X₂' ∩ X₁) + P (X₃ ∩ X₂' ∩ X₁')

Thus, the probability that the third part is the one with excessive shrinkage is 0.2000.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24