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Math  /  Data & Statistics

Question(1 point) Below is a certain probability model: \begin{tabular}{|c|c|c|c|c|c|} \hline Outcome & aa & bb & cc & dd & ee \\ \hline Probability & 0.2 & & 0.15 & 0.05 & 0.3 \\ \hline \end{tabular}
Events FF and GG are given by F={a,c,e},G={b,c,e}.F=\{a, c, e\}, \quad G=\{b, c, e\} .
Calculate each probability: P(b)=0.15P(F∪G)=0.7P(F∩G)=0.35P(F′)=0.45\begin{array}{l} P(b)=0.15 \\ P(F \cup G)=0.7 \\ P(F \cap G)=0.35 \\ P\left(F^{\prime}\right)=0.45 \end{array}

Studdy Solution

STEP 1

1. The probability model is complete, meaning the sum of all probabilities is 1.
2. The probabilities of outcomes bb and dd need to be determined.
3. Events FF and GG are subsets of the sample space.

STEP 2

1. Determine the probability of outcome bb.
2. Verify the probability of the union of events FF and GG.
3. Verify the probability of the intersection of events FF and GG.
4. Verify the probability of the complement of event FF.

STEP 3

Calculate the probability of outcome bb using the total probability:
Given probabilities: P(a)=0.2, P(c)=0.15, P(d)=0.05, P(e)=0.3 P(a) = 0.2, \, P(c) = 0.15, \, P(d) = 0.05, \, P(e) = 0.3
The sum of all probabilities should be 1: P(a)+P(b)+P(c)+P(d)+P(e)=1 P(a) + P(b) + P(c) + P(d) + P(e) = 1
Substitute known values: 0.2+P(b)+0.15+0.05+0.3=1 0.2 + P(b) + 0.15 + 0.05 + 0.3 = 1
P(b)=1−(0.2+0.15+0.05+0.3) P(b) = 1 - (0.2 + 0.15 + 0.05 + 0.3)
P(b)=1−0.7=0.3 P(b) = 1 - 0.7 = 0.3

STEP 4

Verify P(F∪G) P(F \cup G) :
F={a,c,e} F = \{a, c, e\} G={b,c,e} G = \{b, c, e\}
F∪G={a,b,c,e} F \cup G = \{a, b, c, e\}
P(F∪G)=P(a)+P(b)+P(c)+P(e) P(F \cup G) = P(a) + P(b) + P(c) + P(e)
P(F∪G)=0.2+0.3+0.15+0.3=0.95 P(F \cup G) = 0.2 + 0.3 + 0.15 + 0.3 = 0.95
Given P(F∪G)=0.7 P(F \cup G) = 0.7 , there seems to be an inconsistency. Re-evaluate the problem setup or assumptions.

STEP 5

Verify P(F∩G) P(F \cap G) :
F∩G={c,e} F \cap G = \{c, e\}
P(F∩G)=P(c)+P(e)=0.15+0.3=0.45 P(F \cap G) = P(c) + P(e) = 0.15 + 0.3 = 0.45
Given P(F∩G)=0.35 P(F \cap G) = 0.35 , there seems to be an inconsistency. Re-evaluate the problem setup or assumptions.

STEP 6

Verify P(F′) P(F') :
F′={b,d} F' = \{b, d\}
P(F′)=P(b)+P(d)=0.3+0.05=0.35 P(F') = P(b) + P(d) = 0.3 + 0.05 = 0.35
Given P(F′)=0.45 P(F') = 0.45 , there seems to be an inconsistency. Re-evaluate the problem setup or assumptions.
The calculations indicate inconsistencies with the given probabilities. Please verify the problem statement and provided values.

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