Lesson video

Hi, my name's Mr. Chan and in this lesson we're going to learn how to calculate probabilities of dependent events.

Let's begin with this example.

As we can see, we've got a bag containing four pink and three blue counters.

A counter's chosen at random and not replaced.

Then, another counter is chosen at random and we've got to calculate the probability that two pink counters are chosen.

So, I'm going to begin my probability tree diagram with two picks, 'cause we're picking out the counters twice.

And we can pick out pink or blue counters.

And our tree has two branches to pink or blue and we can see that because we have four pink counters out of a total possible number of seven, we have a probability of 4/7 and the blue counters would be 3/7.

Our second pick, again we can only pick out pink or blue, so we have branches leading off to pink or blue.

Now, let's consider if we have picked out our pink one, we would have one less pink counter and one less number of counters all together.

So, it would be 3/6.

And, we can see from the diagram, we also have three blue, that hasn't changed, but we have one less in total, 3/6.

Now, if we'd have picked out a blue counter to start with, our probability of picking out pink would be 4/6 and the probability of picking out blue again would be 2/6.

There's one less blue and one less counter all together.

So, in terms of our outcomes, these are our outcomes of probabilities.

Probability of pink-pink, pink-blue, blue-pink, and finally blue and then blue again.

So, in terms of calculating our probabilities for all of those four outcomes, we look across the branches for pink and then pink and we multiply those values together.

So, we would get 4/7 multiplied by 3/6 to give an answer, 12/42.

Similarly with pink and then blue, we look along the pink first pick branch and then along the second pick along the blue branch, we have 4/7 multiplied by 3/6 to give an answer 12/42.

Similarly with the other outcomes, we're multiplying across the branches to get those answers as you can see there.

Now, we've calculated the probabilities of all of the four outcomes there and what you should notice is that they all add up to equal one, which they do.

So, let's go back to the question.

We're looking for calculating the probability that two pink counters are chosen.

So, that would be this outcome here.

The probability of firstly picking out a pink and then picking out a pink again.

So, our final answer would be the probability of pink and then pink would be 12/42.

If we change the question slightly from the previous example, we are now asked to calculate the probability that one of each colour counter is chosen.

So, one of each colour counter would mean we're going to look for one pink and then one blue or one blue and then one pink.

So these are the two possible outcomes.

Now, when we have more than one possible outcome and we've calculated the probability of all the outcomes, we would add those two probabilities together.

So if we add together 12/42 with 12/42 again, we get a final answer for this question, 24/42.

Now, we could simplify that fraction, but we don't necessarily have to because that probability is accepted as an answer.

Here's another example.

So we've got a bag containing seven red and five green sweets, and Tommy chooses two sweets and he eats them, so calculate the probability that both sweets that he eats are the same colour.

So, there's my bag of sweets that I've just drawn to help us understand how we draw the tree diagram, and Tommy's going to pick two sweets out of the bag.

So, he can only pick red and green, so this is how our tree diagram will begin.

We can see from the first pick, the bag he has seven red out of a total number of 12 sweets, so we have 7/12 in terms of the probability that Tommy can pick out a red sweet.

And, for green he has 5/12, so that would be the probability along that branch.

When Tommy picks out a second sweet, again, he can only pick out red or green.

So, what happens when he picks out a red to start with and he's eaten it? Obviously he's eaten it so he's not going to put it back in the bag.

That means he's got one less red sweet and there's one less sweet all together in the bag.

So now there's 11 sweets in the bag and there are only six red.

So, the probability of a red being picked out in the second pick would be only 6/11 now.

In terms of picking out green, there are still five green sweets, but again, there's one less sweet all together in the bag.

What happens if Tommy had firstly picked out a green sweet then? So, we take the green sweet out, and as you can see there's one less sweet all together.

But more importantly, there's also one less green sweet as well.

So, the probability of red, there's seven there out of 11.

And the probability of green, well he's eaten one of the green sweets so he would have four left, out of 11.

Now, those are our outcomes.

Red and red, red and green, green and red, green and green.

So, in terms of calculating all of the probabilities, we look along all the branches and we would multiply those values together.

So, the probability of red and red would be 7/12 multiplied by 6/11, to give an answer of 42/132.

We do that with all the other outcomes to get those answers there.

And you will notice, the probability of all the outcomes there, again, we just check that they all add up to one, which they do.

And, we can now answer the question.

Calculate the probability that both sweets are the same colour.

Well, "both sweets are the same colour" could indicate that they're both red or both green.

So, we got two possible outcomes there.

Where we have two possible outcomes, we add those probabilities together.

So we add together 42/132 with 20/132 to get an answer of 62/132.

Again, we could simplify that but we don't need to because that answer is acceptable.

Here's a question for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here's the answer.

In this question, we have to work out the probability that Jack takes two red sweets from the bag.

So we use the tree diagram to look at the probabilities of the first sweet being red, 7/10, followed by the second sweet also being red, 6/9.

Those two probabilities we multiply together to get an answer of 42/90, which can then be simplified if we want to, to 7/15.

Here's another question for you to try.

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Here's the answer.

In this question, the probability that the two counters are different colours would indicate that the outcome's must be probability of green and then blue, or the probability of blue and then green.

You got two outcomes, where you work out the probabilities and add those two probabilities together to get the final answer.

Here's another question you can try.

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Resume the video once you're finished.

Here's the answer for question three.

When we're working out the probability that Rosie eats at least one piece of apple, what we have to ask ourselves, "What outcomes are we looking for?" So, the outcomes that would indicate one piece of apple would be probability of apple and then apple, or the probability of apple and then pear, or the probability of a pear first and then an apple.

So there are three outcomes you've got to consider for this answer.

When you've worked out the probability of each of those outcomes, those three outcomes, remember to add them together to get your final answer.

Here's a question for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here are the answers.

With this question, it's really important to figure out how many green marbles there are and how many marbles there are all together in Jim's box.

We can do that from the information that Jim has 24 blue marbles, and the probability of picking out a blue marble is 0.

2.

So, if we know that 0.

2 represents 24 marbles, we can then figure out what the probability of 0.

8 is.

We could simply multiply the value of the number of marbles by four.

So, if we multiply the 24 marbles by four, we get 96 marbles.

So the probability of green being 0.

8 would mean that there are 96 marbles.

And then, the total number of marbles would, in that case, be 120.

So, given that information now, we've got all the information we need to figure out how to draw our probability tree, remember that he's picking out two marbles and giving them away, so he's not replacing them.

So this is indeed dependent and conditional probability.

That's all for this lesson.

Thanks for watching.