A chess club has 5 players with unequal skill. Laurie is the best player. The probability that she will win depends on who her opponent is, with probabilities of winning of .6, .7, .8 and .9, respectively, when playing against members Carrie, David, Paul and Rose. Whether she wins or loses is independent from one game to the next. a. What is the probability that Laurie will lose when she plays against Rose? b. If Laurie plays one game against each player, what is the probability that she does not lose any games? c. If Laurie plays one game against each player, what is the probability that she loses at least once?

a) P(R') = 0.1

b) Probability that she doesn't lose any game if she plays them all = 0.3024

c) If Laurie plays one game against each player, the probability that she loses at least once = 0.6976

Step-by-step explanation:

Let the probability that Laurie will win against Carrie, David, Paul and Rose be P(C), P(D), P(P) and P(R) respectively.

P(C) = 0.6

P(D) = 0.7

P(P) = 0.8

P(R) = 0.9

a) Probability that Laurie will lose when she plays against Rose = P(R') = 1 - P(R) = 1 - 0.9 = 0.1

b) If Laurie plays one game against each player, the probability that she does not lose any games = the probability that she wins all the games = P(C) × P(D) × P(P) × P(R) = 0.6 × 0.7 × 0.8 × 0.9 = 0.3024

c) If Laurie plays one game against each player, the probability that she loses at least once is a sum of a number of probabilities

Take note,

Probability that she wins, against each opponent

P(C) = 0.6

P(D) = 0.7

P(P) = 0.8

P(R) = 0.9

The probability that she loses against each opponent

P(C') = 1 - P(C) = 1 - 0.6 = 0.4

P(D') = 0.3

P(P') = 0.2

P(R') = 0.1

1) Probability that Laurie loses only once while playing the 4 of them

= [P(C') × P(D) × P(P) × P(R)] + [P(C) × P(D') × P(P) × P(R)] + [P(C) × P(D) × P(P') × P(R)] + [P(C) × P(D) × P(P) × P(R')]

= (0.4×0.7×0.8×0.9) + (0.6×0.3×0.8×0.9) + (0.6×0.7×0.2×0.9) + (0.6×0.7×0.8×0.1) = 0.4404

2) Probability that she loses twice playing against each of them

= [P(C') × P(D') × P(P) × P(R)] + [P(C') × P(D) × P(P') × P(R)] + [P(C') × P(D) × P(P) × P(R')] + [P(C) × P(D') × P(P') × P(R)] + [P(C) × P(D') × P(P) × P(R')] + [P(C) × P(D) × P(P') × P(R')]

= (0.4×0.3×0.8×0.9) + (0.4×0.7×0.2×0.9) + (0.4×0.7×0.8×0.1) + (0.6×0.3×0.2×0.9) + (0.6×0.3×0.8×0.1) + (0.6×0.7×0.2×0.1) = 0.2144

3) Probability that Laurie loses three times while playing against them

= [P(C') × P(D') × P(P') × P(R)] + [P(C') × P(D') × P(P) × P(R')] + [P(C') × P(D) × P(P') × P(R')] + [P(C) × P(D') × P(P') × P(R')]

= (0.4×0.3×0.2×0.9) + (0.4×0.3×0.8×0.1) + (0.4×0.7×0.2×0.1) + (0.6×0.3×0.2×0.1) = 0.0404

4) Probability that she loses to the 4 opponents in a game against each one

= [P(C') × P(D') × P(P') × P(R')] = 0.4×0.3×0.2×0.1 = 0.0024

If Laurie plays one game against each player, the probability that she loses at least once = 0.4404 + 0.2144 + 0.0404 + 0.0024 = 0.6976

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: