suppose you flip a fair coin five times. which of these pairs of events are disjoint?

(a). you get five heads;you get four heads and a tail
(b). the first flip is a head;the second flip is a head
(c). you get five heads; the second flip is a tail
(d). you get three heads and two tails;the secon flip is a tail
(e). the first four flips are heads;the first three flips are heads
(F). the first head occurs on the third flip;the fourth flip is a head
(g). the first head occurs on the third flip; three of the flips are heads

Disjoint events are events that cannot occur at the same time.

Let's analyze each pair of events:

(a) You get five heads; you get four heads and a tail
These events are not disjoint as you can get four heads and a tail in both cases.

(b) The first flip is a head; the second flip is a head
These events are disjoint since the first flip being a head does not affect the outcome of the second flip.

(c) You get five heads; the second flip is a tail
These events are disjoint as getting five heads means all flips are heads, while the second flip being a tail means at least one flip is not a head.

(d) You get three heads and two tails; the second flip is a tail
These events are not disjoint as the second flip being a tail can happen in both cases.

(e) The first four flips are heads; the first three flips are heads
These events are not disjoint as the first three flips being heads is a subset of the first four flips being heads.

(f) The first head occurs on the third flip; the fourth flip is a head
These events are disjoint since the fourth flip being a head means there was already a head on the third flip.

(g) The first head occurs on the third flip; three of the flips are heads
These events are not disjoint as three of the flips being heads can happen in both cases.

In summary, the pairs of disjoint events are:
(b) The first flip is a head; the second flip is a head
(c) You get five heads; the second flip is a tail
(f) The first head occurs on the third flip; the fourth flip is a head

To determine if a pair of events is disjoint, we need to check if they can occur at the same time or not. In other words, if one event happening means that the other event cannot happen, then they are disjoint.

Let's analyze each pair of events:

(a). You get five heads; you get four heads and a tail.
These two events cannot occur at the same time because if you get five heads, it means you cannot have a tail. Therefore, these events are disjoint.

(b). The first flip is a head; the second flip is a head.
These events can occur at the same time. The first flip being a head does not prevent the second flip from being a head. Therefore, these events are not disjoint.

(c). You get five heads; the second flip is a tail.
These two events cannot occur at the same time because if you get five heads, it means you cannot have a tail. Therefore, these events are disjoint.

(d). You get three heads and two tails; the second flip is a tail.
These events can occur at the same time. Getting three heads and two tails in five flips does not prevent the second flip from being a tail. Therefore, these events are not disjoint.

(e). The first four flips are heads; the first three flips are heads.
These two events can occur at the same time. If the first four flips are heads, it also means that the first three flips are heads. Therefore, these events are not disjoint.

(f). The first head occurs on the third flip; the fourth flip is a head.
These two events can occur at the same time. The first head occurring on the third flip does not prevent the fourth flip from being a head. Therefore, these events are not disjoint.

(g). The first head occurs on the third flip; three of the flips are heads.
These two events can occur at the same time. If the first head occurs on the third flip, it is still possible for three of the flips to be heads. Therefore, these events are not disjoint.

Based on our analysis, the pairs of events that are disjoint are:
(a). You get five heads; you get four heads and a tail.
(c). You get five heads; the second flip is a tail.

Which events are mutually exclusive — they cannot occur at the same time?

I hope this helps you to make your decision.