URGENT HELP NEEDED ASAP FOR STATISTICS

Question

suppose a certain candy is made up from red, yellow, green, blue and orange pieces.

a. is you draw one piece of candy, are the events of getting a yellow one and getting a blue one disjoint, independent or neither?

b. assume you an infinite supply of these candy pieces from which to draw. if you draw two pieces one right after the other, are the events of getting a green one on the fist and a green on the second disjoint, independent or neither?

c. can disjoint events ever be independent? explain.

Answer 1

a. Disjoint refers to events that never occur at same time, so disjoint is not applicable in the case of drawing one piece of candy.

Independent refers to unrelated events. Independent is the correct answer because the probability of drawing a yellow or drawing a blue are entirely unrelated to each other, given we’re talking about one draw.

b. Disjoint refers to events that never occur at the same time. It is possible to draw 2 green pieces [one right after the other] from the infinite supply, and therefore this example is joint [and therefore not disjoint].

Independent refers to unrelated events. Independent is the correct answer because the probability of the 2^nd piece being green is entirely unrelated to the color of the 1^st piece drawn.

Please note, when an object is sampled with replacement, the outcome of the next event is entirely unrelated to the outcome of the prior event; however, when an object is sampled without replacement,  the outcome of the next event is related to the outcome of the prior event. Given that an infinite supply is effectively the same as sampling with replacement, we conclude the 2^nd piece drawn being green is entirely unrelated to the outcome of the 1^st piece drawn being green.

c. Disjoint events and independent events are easily confused concepts. Short answer is no, a disjoint event can never be considered independent; in fact, quite the opposite. Think about it like this – since disjoint events can never occur at the same time, if event 1 occurs and event 2 is disjoint with event 1, then event 2 is very much dependent on event 1. In fact, given that event 1 occurred, we now know event 2 will not occur, specifically because event 2 is disjoint with event 1. In summary, easy to confuse disjoint with independent, but the entire premise behind disjoint events is that they are in fact dependent, precisely because they cannot occur at the same time.

As another example to illustrate the point of disjoint being dependent. Imagine it is nighttime in this precise location at this precise moment. Can it also be daytime in this precise location at this precise moment? Of course not, so these are disjoint events. In fact, it not being daytime in this precise location at this precise moment is very much dependent on the fact that it is nighttime at this precise location at this precise moment.