91Ó°ÊÓ

A construction company is bidding on three projects: \(B_{1}, B_{2}\) and \(B_{3}\). From previous experience they have the following probabilities of winning the bids: \(P\left(B_{1}\right)=0.22, P\left(B_{2}\right)=0.25\) and \(P\left(B_{3}\right)=0.28 .\) Winning the bids is not independent one from another. The joint probabilities are given below. $$\begin{array}{|c|c|c|c|} \hline & B_{1} & B_{2} & B_{3} \\ \hline B_{1} & & 0.11 & 0.05 \\ \hline B_{2} & 0.11 & & 0.07 \\ \hline B_{3} & 0.05 & 0.07 & \\ \hline \end{array}$$ Also, \(P\left(B_{1} \cap B_{2} \cap B_{3}\right)=0.01 .\) Find the following probabilities: a) \(P\left(B_{1} \cup B_{2}\right)\) b) \(P\left(B^{\prime}_{1} \cap B_{2}^{\prime}\right)\) c) \(P\left(B^{\prime}, \cap B_{2}^{\prime}\right) \cup B_{3}\) d) \(P\left(B^{\prime}_{1} \cap B_{2}^{\prime} \cap B_{3}\right)\) e) \(P\left(B_{2} \cap B_{3} | B_{1}\right)\) \(f) P\left(B_{2} \cup B_{3} | B_{1}\right)\)

Step by step solution

01

Calculate P(B1 ∪ B2)

Use the principle of inclusion-exclusion to find the union probability: \[ P(B_1 \cup B_2) = P(B_1) + P(B_2) - P(B_1 \cap B_2) \]Substitute the given values:\[ P(B_1 \cup B_2) = 0.22 + 0.25 - 0.11 = 0.36 \]

02

Calculate P(B1' ∩ B2')

Use the complement rule:\[ P(B_1' \cap B_2') = 1 - P(B_1 \cup B_2) \]Substitute the value from Step 1:\[ P(B_1' \cap B_2') = 1 - 0.36 = 0.64 \]

03

Calculate P((B1' ∩ B2') ∪ B3)

Apply the rules of probability:\[ P((B_1' \cap B_2') \cup B_3) = P(B_1' \cap B_2') + P(B_3) - P((B_1' \cap B_2') \cap B_3) \]Calculate using known probabilities:\[ P((B_1' \cap B_2') \cap B_3) = P(B_3) - P(B_1 \cap B_3) - P(B_2 \cap B_3) + P(B_1 \cap B_2 \cap B_3) \]Therefore,\[ P((B_1' \cap B_2') \cap B_3) = 0.28 - 0.05 - 0.07 + 0.01 = 0.17 \]Finally, substitute back:\[ P((B_1' \cap B_2') \cup B_3) = 0.64 + 0.28 - 0.17 = 0.75 \]

04

Calculate P(B1' ∩ B2' ∩ B3)

Use conditional probabilities:Previously calculated \[ P((B_1' \cap B_2') \cap B_3) = 0.17 \]

05

Calculate P(B2 ∩ B3 | B1)

By the conditional probability formula:\[ P(B_2 \cap B_3 | B_1) = \frac{P(B_1 \cap B_2 \cap B_3)}{P(B_1)} \]Substituting given values:\[ P(B_2 \cap B_3 | B_1) = \frac{0.01}{0.22} \approx 0.0455 \]

06

Calculate P(B2 ∪ B3 | B1)

Use the conditional probability formula:\[ P(B_2 \cup B_3 | B_1) = \frac{P(B_2 \cup B_3 \cap B_1)}{P(B_1)} \]Substitute previous values:\[ P(B_2 \cup B_3 \cap B_1) = P(B_2 \cap B_1) + P(B_3 \cap B_1) - P(B_2 \cap B_3 \cap B_1) \]\[ P(B_2 \cup B_3 \cap B_1) = 0.11 + 0.05 - 0.01 = 0.15 \]Thus,\[ P(B_2 \cup B_3 | B_1) = \frac{0.15}{0.22} \approx 0.6818 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It's one of the fundamental concepts in probability theory. For example, if we want to find the probability of winning project B2 and B3 given that project B1 has been won, we use the conditional probability formula:\[P(B_2 \cap B_3 | B_1) = \frac{P(B_1 \cap B_2 \cap B_3)}{P(B_1)}\]This formula helps us understand how probabilities change when we already have information about the first event. It's similar to narrowing our focus based on new, relevant information.
Let's take an example. Imagine we're flipping two coins. The probability of getting two heads given that the first flip is a head, shifts our perspective and probability calculations.
Key points in conditional probability involve:

• Understanding how new information changes likelihoods.
• Knowing that the scenario becomes limited to the conditions set.
• Applying this in real-life contexts, such as analyzing risks or outcomes in dependent situations.

Joint Probability

Joint probability is all about finding the likelihood of two or more events occurring at the same time. It's essential when events are interconnected. In our case of bidding on construction projects, joint probability helps us determine how often the success of different bids happens together.
For instance, we can see the joint probability through values like:

• \(P(B_1 \cap B_2) = 0.11\)
• \(P(B_2 \cap B_3) = 0.07\)

These figures highlight the chance of the specific combinations succeeding in tandem.
To compute joint probabilities, two major scenarios exist:

• If events are independent, their joint probability is simply the product of their individual probabilities.
• If not, the given provided dependencies (like bid interrelations) need direct calculations or additional information.

Joint probabilities provide insights into overlaps and combinations, crucial in areas like cross-category marketing or concurrent outcomes predictions.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a handy tool for finding probabilities involving unions of events. It helps when calculating the probability of either one event or another occurring. In our exercise, we used it to find the probability of winning at least one of two projects, B1 or B2, for instance.
The inclusion-exclusion principle formula is straightforward:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]This formula ensures we account for the overlap we counted twice when simply adding probabilities for both events.
Here is how it works:

• First, sum the probabilities of each separate event.
• Then, subtract the probability of both events happening together, which was counted twice.

The principle extends beyond two events. It systematically adjusts for multiple overlaps and is key in scenarios, like survey interpretation or network analysis, where overlapping situations are typical.

Complement Rule

The complement rule provides an efficient way to find the probability of something not happening. It's often simpler than directly calculating the probability of the event itself. This rule is valuable because it shifts the focus to what's left after an event happens.
The complement rule is expressed as:\[P(A') = 1 - P(A)\]Where \(A'\) is the event not happening, and \(P(A)\) is the probability of the event happening.
In our problem scenario, to find the probability that both bids B1 and B2 are lost, we used:\[P(B_1' \cap B_2') = 1 - P(B_1 \cup B_2)\]It's often more intuitive to think of complements in scenarios like spinning a wheel. If the wheel lands on black 40% of the time, it must land on other colors the remaining 60% of the time.
The complement rule is essential for calculating odds in many areas, such as reliability in engineering or determining market shares in business.