Problem: 5 coins are tossed, find the probability of getting a) Exact 3 heads b) At least 4 heads c) At most 2 heads?

Solution:

When a coin is tossed, there are 2 possible outcomes - head or tail. Therefore, when 5 coins are tossed, there are 2^5 = 32 possible outcomes.

Let's denote head as H and tail as T. Then the possible outcomes when 5 coins are tossed are:

HHHHH, HHHHT, HHHTH, HHHTT, HHTHH, HHTHT, HHTTH, HHTTT, HTHHH, HTHHT, HTHTH, HTHTT, HTTHH, HTTHT, HTTTH, HTTTT, THHHH, THHHT, THHTH, THHTT, THTHH, THTHT, THTTH, THTTT, TTHHH, TTHHT, TTHTH, TTHTT, TTTHH, TTTHT, TTTTH, TTTTT

Let's now solve the given problems one by one:

Exact 3 heads:
To get exactly 3 heads, we need to count the number of outcomes where there are exactly 3 heads. From the possible outcomes listed above, we can see that there are 10 outcomes where there are exactly 3 heads. Therefore, the probability of getting exactly 3 heads is:

P(exactly 3 heads) = Number of outcomes with exactly 3 heads / Total number of possible outcomes
= 10/32
= 5/16

At least 4 heads:
To get at least 4 heads, we need to count the number of outcomes where there are 4 or 5 heads. From the possible outcomes listed above, we can see that there are 5 outcomes where there are 4 heads and 1 outcome where there are 5 heads. Therefore, the probability of getting at least 4 heads is:

P(at least 4 heads) = Number of outcomes with 4 or 5 heads / Total number of possible outcomes
= (5+1)/32
= 3/16

At most 2 heads:
To get at most 2 heads, we need to count the number of outcomes where there are 0, 1, or 2 heads. From the possible outcomes listed above, we can see that there are 1 outcome where there are 0 heads, 5 outcomes where there is 1 head, and 10 outcomes where there are 2 heads. Therefore, the probability of getting at most 2 heads is:

P(at most 2 heads) = Number of outcomes with 0, 1, or 2 heads / Total number of possible outcomes
= (1+5+10)/32
= 16/32
= 1/2

Therefore, the probability of getting a) exact 3 heads is 5/16 b) at least 4 heads is 3/16 and c) at most 2 heads is 1/2.