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PROBLEM

Part 6 of 6
Points: 0%0 \%, 9 of 10 points
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Color blindness is an inherited characteristic that is more common in males than in females. If M represents male and C represents red-green color blindness, using the relative frequencies of the incidences of males and red-green color blindness as probabilities results in the values below. Complete parts (a) through ( ff ) below.
P(C)=0.035,P(M∩C)=0.033,P(M∪C)=0.487P(C)=0.035, P(M \cap C)=0.033, P(M \cup C)=0.487 (a) Find P(C′)P\left(C^{\prime}\right).
P(C′)=0.965P\left(C^{\prime}\right)=0.965 (Type an integer or a decimal.)
(b) Find P(M)P(M).
P(M)=0.485P(M)=0.485 (Type an integer or a decimal.)
(c) Find P(M′)P\left(M^{\prime}\right).
P(M′)=0.515P\left(M^{\prime}\right)=0.515 (Type an integer or a decimal.)
(d) Find P(M′∩C′)P\left(M^{\prime} \cap C^{\prime}\right).
P(M′∩C′)=0.513P\left(M^{\prime} \cap C^{\prime}\right)=0.513 (Type an integer or a decimal.)
(e) Find P(C∩M′)P\left(C \cap M^{\prime}\right).
P(C∩M′)=0.002P\left(C \cap M^{\prime}\right)=0.002 (Type an integer or a decimal.)
(f) Find P(C∪M′)P\left(C \cup M^{\prime}\right).
P(C∪M′)=□4P\left(C \cup M^{\prime}\right)=\square_{4} (Type an integer or a decimal.)

STEP 1

1. We are given the probabilities P(C)=0.035 P(C) = 0.035 , P(M∩C)=0.033 P(M \cap C) = 0.033 , and P(M∪C)=0.487 P(M \cup C) = 0.487 .
2. We need to find the probabilities of complementary and combined events using the given data.

STEP 2

1. Calculate P(C′) P(C') .
2. Calculate P(M) P(M) .
3. Calculate P(M′) P(M') .
4. Calculate P(M′∩C′) P(M' \cap C') .
5. Calculate P(C∩M′) P(C \cap M') .
6. Calculate P(C∪M′) P(C \cup M') .

STEP 3

Calculate the probability of the complement of C C , P(C′) P(C') :
P(C′)=1−P(C)=1−0.035=0.965 P(C') = 1 - P(C) = 1 - 0.035 = 0.965

STEP 4

Calculate the probability of M M using the formula for the union of two events:
P(M∪C)=P(M)+P(C)−P(M∩C) P(M \cup C) = P(M) + P(C) - P(M \cap C) Rearrange to solve for P(M) P(M) :
P(M)=P(M∪C)−P(C)+P(M∩C) P(M) = P(M \cup C) - P(C) + P(M \cap C) P(M)=0.487−0.035+0.033=0.485 P(M) = 0.487 - 0.035 + 0.033 = 0.485

STEP 5

Calculate the probability of the complement of M M , P(M′) P(M') :
P(M′)=1−P(M)=1−0.485=0.515 P(M') = 1 - P(M) = 1 - 0.485 = 0.515

STEP 6

Calculate the probability of the intersection of the complements, P(M′∩C′) P(M' \cap C') :
Using the formula for the complement of the union:
P(M′∩C′)=1−P(M∪C) P(M' \cap C') = 1 - P(M \cup C) P(M′∩C′)=1−0.487=0.513 P(M' \cap C') = 1 - 0.487 = 0.513

STEP 7

Calculate the probability of C∩M′ C \cap M' :
Using the formula for the intersection of complements:
P(C∩M′)=P(C)−P(M∩C) P(C \cap M') = P(C) - P(M \cap C) P(C∩M′)=0.035−0.033=0.002 P(C \cap M') = 0.035 - 0.033 = 0.002

SOLUTION

Calculate the probability of the union of C C and M′ M' :
Using the formula for the union:
P(C∪M′)=P(C)+P(M′)−P(C∩M′) P(C \cup M') = P(C) + P(M') - P(C \cap M') P(C∪M′)=0.035+0.515−0.002=0.548 P(C \cup M') = 0.035 + 0.515 - 0.002 = 0.548 The probability P(C∪M′) P(C \cup M') is:
0.548 \boxed{0.548}

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