Sample Spaces and Events

Transcript

0:00Introduction

Hello everyone! Let me start with a small puzzle. You toss a coin three times. How many different results can you get? Many students say four: zero, one, two or three heads. But is that really the full list? Keep that question in mind, because we will settle it in the first example. Every probability question starts with the sample space and its events, and they come up often in JEE, so let's begin.

0:29Experiment, sample space, event

So, what is a random experiment? It has several possible outcomes, and we cannot say in advance which one will occur. Okay? Now, collect all the possible outcomes into one set. That set is the sample space, written as capital S, right? And what is an event? Any subset of S is an event: any collection of outcomes from the sample space. Clear? Let's make it real. Throw one die. The sample space S is the set one, two, three, four, five, six. The curly brackets mean a set, okay? Now tell me, which outcomes make the event, an even number? Two, four and six, yes, exactly! So E is the set two, four, six. This sign, like a rounded less than or equal to, means is a subset of. Every event lives inside the sample space, got it?

1:19Or, and, not: De Morgan's laws

Now, events are sets, so we can combine them. In the left picture, the rectangle marked S is the sample space, and the two circles are events A and B, okay? The event, A or B, means at least one of them happens. We write it as A union B, with this cup shaped symbol. The event, A and B, means both happen together. That is A intersection B, the upside down cup: the darker overlap, where the arrow points. Right? Not A is written A prime, with a small dash called a prime. It is the complement of A: every outcome of S that lies outside A, okay? So in the picture, A intersection B prime is the part of A that lies outside B, with me? Now let me ask you this: how do you write the event, neither A nor B? Neither means not A and not B, so it is A prime intersection B prime, very good! That is also the complement of A union B, because the prime outside the bracket means the complement of everything inside. This is De Morgan's law, clear? The other De Morgan's law: the complement of A intersection B equals A prime union B prime, which means not both. So at most one of A and B is the complement of A intersection B, okay? Careful, students often write exactly one here, but at most one also allows neither. They are not the same.

2:38Mutually exclusive and exhaustive

Next, two important words. Events A and B are mutually exclusive when A intersection B is the empty set, this circle with a slash through it. See, the two circles do not overlap, so both can never happen together, right? And events are exhaustive when their union, the shaded part, is the whole sample space S, right? So for exhaustive events, the white region outside both circles must be empty: no outcome is left out. The first word is about the overlap, the second is about covering everything. Two different ideas, okay? Now one very useful split. Event A breaks into two pieces: the lighter middle part, A intersection B, and the darker part, A intersection B prime. So A equals A intersection B, union A intersection B prime. The two pieces never overlap, so they are mutually exclusive. Clear?

3:29Exactly one of the events

Look at the left picture again. Which parts of it make up the event, exactly one of A and B? The two outer parts, yes, exactly! So exactly one of them is A intersection B prime, union A prime intersection B, okay? Not the middle, because there both of them happen. Now three events, in the right picture. Letters written side by side mean intersection, so A B prime C prime means A, and not B, and not C, right? And the shaded core, marked ABC, is where all three happen. Now, using this short notation, how would you write the event, exactly one of A, B and C? Three pieces joined by union, yes, exactly! It is A B prime C prime, union A prime B C prime, union A prime B prime C, okay? Look at the pattern: in each piece, exactly one letter has no prime. Really simple, isn't it?

4:27Example 1: coin tossed three times, exactly two heads

Now, remember our puzzle from the start? Let's settle it. So, a coin is tossed three times. Write the sample space, and the event, exactly two heads. Pause the video and try it first. Done? Okay, let's check. We write H for a head and T for a tail. Each toss has two results, so we get two times two times two. That is two to the power three, the small three up top, which gives eight outcomes. Here they are: HHH, then the three with two heads, then the three with one head, and TTT at the end. Eight outcomes, not four! Zero, one, two or three heads was only a count of heads, and one count can cover several outcomes, okay? Now look at the list and tell me, how many outcomes have exactly two heads? Three of them, very good! HHT, HTH and THH, right? The tail can come on any one of the three tosses, and these are different outcomes. Writing just one of them is the usual slip. So the event E has three outcomes, out of the eight in the sample space. That's it, puzzle solved!

5:36Example 2: two dice, sum 7 vs double, exclusive? exhaustive?

Now a slightly harder one. Two dice are thrown. Are the events, sum is seven, and, a double, mutually exclusive? Are they exhaustive? Pause the video and try it first. Done? Okay, let's check. Each die has six faces, so the sample space has six times six, that is thirty six outcomes. Clear? Let A be the event, sum is seven. Each outcome is an ordered pair in round brackets, first die, then second die. One and six, two and five, three and four, and these three reversed: six outcomes, shown as six in the first circle. And one, six is different from six, one, okay? Let B be the event, a double: one and one, two and two, up to six and six. The three dots mean the pattern carries on. Again six outcomes, in the second circle, with me? Now the key question. Can a double ever have a sum of seven? No, never, very good! The sum of a double is a number plus itself, so it is always even, and seven is odd. So A intersection B is the empty set, and that is why the circles do not touch. They are mutually exclusive, right? So they are mutually exclusive. Does that make them exhaustive too? No, no, careful here, because these are two different ideas. Together the events hold six plus six, only twelve outcomes, and twelve is less than thirty six. So the answer is mutually exclusive, but not exhaustive. Got it?

7:03Quick revision: events, exclusive vs exhaustive, De Morgan

So, quick revision. The sample space S is the set of all outcomes, and an event is any subset of it. Mutually exclusive means no overlap, exhaustive means the union is all of S, and one does not give you the other. Neither is A prime intersection B prime, and at most one is the complement of A intersection B, okay? Now go and try the practice problem on this page. All the best!