Math / Data & Statistics

QuestionThe accompanying tree diagram represents an experiment consisting of two trials.
Use the diagram to find the probabilities below. $\begin{array}{l} \text { (a) } P(A) \\ 1.4 \end{array}$ $\begin{array}{l} \text { (b) } \quad P(E \mid A) \\ 5 \\ \text { (c) } \quad P(A \cap E) \\ 15 \end{array}$ (d) $P(E)$ $35$

Studdy Solution

STEP 1

1. The tree diagram represents a probability experiment with two trials.
2. Probabilities for the first trial are given as $P(A) = 0.4$ , $P(B) = 0.3$ , and $P(C) = 0.2$ .
3. Conditional probabilities for the second trial are given for each branch.

STEP 2

1. Verify and calculate $P(A)$ .
2. Calculate $P(E \mid A)$ .
3. Calculate $P(A \cap E)$ .
4. Calculate $P(E)$ .

STEP 3

Verify the probability of event $A$ .
Given: $P(A) = 0.4$
Since the probability is directly provided in the tree diagram, no further calculation is needed.

STEP 4

Calculate the conditional probability $P(E \mid A)$ .
From the tree diagram, the probability of $E$ given $A$ is:
$P(E \mid A) = 0.5$

STEP 5

Calculate the joint probability $P(A \cap E)$ .
Using the formula for joint probability:
$P(A \cap E) = P(A) \times P(E \mid A)$
Substitute the known values:
$P(A \cap E) = 0.4 \times 0.5 = 0.2$

STEP 6

Calculate the probability of $E$ , $P(E)$ .
Using the law of total probability:
$P(E) = P(A \cap E) + P(B \cap E) + P(C \cap E)$
Calculate each term:
$P(B \cap E) = P(B) \times P(E \mid B) = 0.3 \times 0.3 = 0.09$
$P(C \cap E) = P(C) \times P(E \mid C) = 0.2 \times 0.4 = 0.08$
Add the probabilities:
$P(E) = 0.2 + 0.09 + 0.08 = 0.37$
The probabilities are: (a) $P(A) = 0.4$ (b) $P(E \mid A) = 0.5$ (c) $P(A \cap E) = 0.2$ (d) $P(E) = 0.37$

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Studdy Solution

STEP 1

1. The tree diagram represents a probability experiment with two trials.
2. Probabilities for the first trial are given as $P(A) = 0.4$ , $P(B) = 0.3$ , and $P(C) = 0.2$ .
3. Conditional probabilities for the second trial are given for each branch.

STEP 2

1. Verify and calculate $P(A)$ .
2. Calculate $P(E \mid A)$ .
3. Calculate $P(A \cap E)$ .
4. Calculate $P(E)$ .

STEP 3

Verify the probability of event $A$ .
Given: $P(A) = 0.4$
Since the probability is directly provided in the tree diagram, no further calculation is needed.

STEP 4

Calculate the conditional probability $P(E \mid A)$ .
From the tree diagram, the probability of $E$ given $A$ is:
$P(E \mid A) = 0.5$

STEP 5

Calculate the joint probability $P(A \cap E)$ .
Using the formula for joint probability:
$P(A \cap E) = P(A) \times P(E \mid A)$
Substitute the known values:
$P(A \cap E) = 0.4 \times 0.5 = 0.2$

STEP 6

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Studdy Solution

STEP 1

1. The tree diagram represents a probability experiment with two trials.2. Probabilities for the first trial are given as P(A)=0.4 P(A) = 0.4 P(A)=0.4, P(B)=0.3 P(B) = 0.3 P(B)=0.3, and P(C)=0.2 P(C) = 0.2 P(C)=0.2.3. Conditional probabilities for the second trial are given for each branch.

STEP 2

1. Verify and calculate P(A) P(A) P(A).2. Calculate P(E∣A) P(E \mid A) P(E∣A).3. Calculate P(A∩E) P(A \cap E) P(A∩E).4. Calculate P(E) P(E) P(E).

STEP 3

Verify the probability of event A A A.Given: P(A)=0.4 P(A) = 0.4 P(A)=0.4Since the probability is directly provided in the tree diagram, no further calculation is needed.

STEP 4

Calculate the conditional probability P(E∣A) P(E \mid A) P(E∣A).From the tree diagram, the probability of E E E given A A A is:P(E∣A)=0.5 P(E \mid A) = 0.5 P(E∣A)=0.5

STEP 5

Calculate the joint probability P(A∩E) P(A \cap E) P(A∩E).Using the formula for joint probability:P(A∩E)=P(A)×P(E∣A) P(A \cap E) = P(A) \times P(E \mid A) P(A∩E)=P(A)×P(E∣A)Substitute the known values:P(A∩E)=0.4×0.5=0.2 P(A \cap E) = 0.4 \times 0.5 = 0.2 P(A∩E)=0.4×0.5=0.2

STEP 6

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1. The tree diagram represents a probability experiment with two trials.
2. Probabilities for the first trial are given as $P(A) = 0.4$ , $P(B) = 0.3$ , and $P(C) = 0.2$ .
3. Conditional probabilities for the second trial are given for each branch.

1. Verify and calculate $P(A)$ .
2. Calculate $P(E \mid A)$ .
3. Calculate $P(A \cap E)$ .
4. Calculate $P(E)$ .

Verify the probability of event $A$ .
Given: $P(A) = 0.4$
Since the probability is directly provided in the tree diagram, no further calculation is needed.

Calculate the conditional probability $P(E \mid A)$ .
From the tree diagram, the probability of $E$ given $A$ is:
$P(E \mid A) = 0.5$

Calculate the joint probability $P(A \cap E)$ .
Using the formula for joint probability:
$P(A \cap E) = P(A) \times P(E \mid A)$
Substitute the known values:
$P(A \cap E) = 0.4 \times 0.5 = 0.2$