Gain clarity on key probability concepts such as independent versus dependent events, expected values, and conditional probability. Understand how these fundamental principles apply to real-world scenarios and business decision-making.
Key Insights
- Calculating probabilities with replacement (independent events) results in consistent probabilities, while probabilities without replacement (dependent events) change after each selection due to adjusted conditions.
- Analyzing expected value involves multiplying each outcome by its probability and summing the results, as demonstrated in the case of a potential $500 sale with a 60% probability and a $200 loss with a 40% probability, resulting in a net expected gain of $220.
- "Fair price" refers to the combination of the actual cost and the perceived value customers attribute to a product, exemplified by determining the fair price of a coffee costing $2 to produce, but valued at $4 due to branding considerations.
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Probability 2. We're going to take a look at a couple of more probability exercises. We're going to go back to working with items that we need to pick out of a bag. So for exercise one, we want to determine the probability of someone randomly selecting a red toy and a blue toy on two consecutive attempts with replacement.
Now the term with replacement means whatever ball we pick up, we're going to put back into the bag. So it'll be available in the same pool of balls to pick from for the next attempt. So this specifically says red toy and blue toy.
That means the first toy is going to be red and the second toy has to be blue. It cannot be blue and red. It has to be red first and then blue second.
Now selecting red on the first try, then selecting blue on the second try, there's a very slim chance of getting an exact two combo pairing because you're basically controlling the order and there's no flexibility. You can't say, well, it still qualifies if I pick up blue and then red because it's red and blue. No, it has to be red on the first try and blue on the second.
So in terms of calculating the percentage, picking the red ball on the first try is 20%. Now on the second try, it's still going to be 20% to pick up the blue, but because they need to be in consecutive order, instead of just adding 20% to 20%, we need to multiply. We're multiplying 20% by 20% and the result is 4%.
So there's a 4% chance that we'll pick up a red toy on the first try and then a blue toy on the second try. Now, if you do the exercise over again, but this time use the or statement, well, then you have a little bit more flexibility because you could pick up the blue toy first and then the red toy or the red toy first and then the blue toy. Either way, you're selecting red and blue, but you have flexibility with the order.
So that is also going to be 20%, but instead of multiplying, you're going to add 20% for blue on the second try. So you still have equal possibility, but instead of multiplying, you're going to add. When I press ENTER, I have 40%.
So that is much better in terms of odds and probability. But what I need to do is also take into account the fact that I could pick another pairing or another ball. And so there's a 4% chance that I could pick up a ball that is not a red or blue.
So I'm going to minus 4%. And when I press ENTER, my actual odds are 36%. We have a table on the right that shows you a breakdown of the color combinations for a bag that has five colored balls and the different pairings.
This is very similar to what we did when we're working with the table that showed your odds of picking numbers from craps, the number of dice where you might either get like a 7 or 11 and so on. Let's take a look at a different type of exercise. This one's a little bit more straightforward.
We're going to take a look at whether events are independent or dependent. Now, with replacement means that they're independent because we're setting up the same conditions as our first attempt. So events A and B are independent if the occurrence of one event in no way affects the occurrence of the other.
With replacement means there is no adjustment to the probability of the next event. So there is a basket of five colored toys, three green and two blue. What is the probability of randomly selecting a green toy on two consecutive tries with replacement? So that's going to be equal to 3 divided by 5 because you have three opportunities out of five to pick a green toy.
I'll press ENTER. And that is 60%. Now, when I pick up that green toy, I'm going to put it back in the bag for my second attempt.
So what is the probability on the second attempt? Well, that's simply going to be equal to 3 again divided by 5. And so that's 60%. We're looking at individual events and it's 60% both times. Now, events A and B for dependent events are dependent if the occurrence of one event affects the occurrence of another.
So without replacement means that there is an adjustment to the probability of the next event. So the fact that I am not putting the green ball back reduces my odds because I will no longer have three green balls to pick from. So this is going to be equal to, on the first attempt, 3 divided by 5, which again is 60%.
Same as the last two attempts in the last exercise. But what if I don't put the ball that I picked up back into the bag? Well, now I no longer have three green toys. I have two and two.
So that's going to be equal to 2 divided by 4. That gives me 50% in terms of probability. So that reduces the probability of picking up a green toy. Now my odds are 50-50.
So that's the difference between independent and dependent events. Now we're going to take a look at another type of probability. An executive finishes a business meeting with the following outcomes in mind.
So he would like to complete the purchase. But looking at the odds of completing this purchase, completing the sale, actually, looking at the odds of completing the sale, there's a 60% chance that the client will purchase. And so the benefit that that salesperson has of completing that sale is $500, a sale of $500.
Now, there is a 40% chance that the client will not purchase. That will be a loss of $200. Well, how is that a loss of $200? Well, we need to take into account everything that was involved in setting up the meeting.
That might include travel, dinner, any expenses involved in setting up and completing the meeting. So what we want to find out is what is the expected gain or loss? If we take into account 60% and $500 as a benefit and 40% and 200, how would we weigh the expected gain or loss in total? So this is how you would calculate this. The expected gain of loss would be equal to 60% multiplied by the outcome, which is 500.
Then what we'd like to do is add 40%, which is our chance of not making the sale, multiplied by negative 200. And that's the loss. So I'll press ENTER.
That is $220. So that is my cost benefit analysis. So $200 is the expected gain or loss.
That might give me a sense of whether or not I would actually like to take a chance at making the sale or not. Now, if you change the percentages and change the amounts, of course, it'll affect the expected gain or loss. So you could play around with the percentages and the amounts and see how that affects the expected gain or loss.
Fair price is pretty simple. The sum of the expected value and the required cost, that is what's going to be referred to as the fair price. So I have a coffee cup here.
It costs me, as someone who runs Cafe Joe's, $2 to make the coffee. Now, I have to think about what are people willing to pay for that coffee. If I have a Starbucks logo on my cup, that potentially adds to the value of the coffee because Starbucks is a recognized name brand.
So maybe that is something that increases the expected value. My actual cost is only $2. If I think there's an expected value of $4, then my fair price is going to be equal to the cost, which is $2, plus the expected value, which is $4.
And that's what I will be able to charge for the coffee, and that is what people will be willing to pay. So that's the fair price. Last exercise, we're going to take a look at the probability of one event occurring and how that's affected by occurrence of another event.
We have the mathematical calculation here. The situation is, we want to find out how many customers are satisfied with the new Microsoft Excel. We need to do some calculations based on the results that we get.
So there's a total of 820,000 people who were surveyed, 600,000 were satisfied, 220,000 were dissatisfied. So I need to come up with the percentages of satisfied versus dissatisfied. So those satisfied with Excel are going to be equal to the satisfied amount divided by the total number.
That is going to give me my percentages satisfied. That is 73%. As we learned earlier with a fair game, all the values need to add up to 100%.
So you can probably figure out what the satisfied value is going to be, but let's calculate it anyway. That's going to be equal to the dissatisfied amount divided by the total. I'll press ENTER.
That is 27%. Taken all together, that is 100%. So that is the conditional probability.
The amount of people satisfied impact the amount of people dissatisfied because everything, when it comes to our percentages, needs to add up to 100%. All right. So in this section, we took a look at some more probability exercises.
We looked at independent versus dependent events. We also looked at an expected value, fair price, and conditional probability. Thank you for watching.