A coin is tossed 3 times. Find the probability of obtaining the following:

a) heads on the first and last toss, and tails on the other toss
b) at least 2 heads
c) at most 2 heads

To find the probabilities, we need to consider the sample space, which is the set of all possible outcomes of the coin tosses. Since each toss has two possible outcomes (heads or tails), the sample space for 3 tosses is given by:

Sample Space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

a) Probability of obtaining heads on the first and last toss, and tails on the other toss:
In this case, we are looking for outcomes where the first and last tosses are heads and the middle toss is tails. There is only one outcome that satisfies this condition: HTH. Therefore, the probability is 1/8.

b) Probability of obtaining at least 2 heads:
To find this probability, we need to count all the outcomes where we get 2 or 3 heads. Looking at the sample space, the outcomes that satisfy this condition are: HHH, HHT, HTH, THH. There are 4 such outcomes, so the probability is 4/8 or 1/2.

c) Probability of obtaining at most 2 heads:
To find this probability, we need to count all the outcomes where we get 0, 1, or 2 heads. Looking at the sample space again, the outcomes that satisfy this condition are: TTT, TTH, THT, HTT, HHT, THH. There are 6 such outcomes, so the probability is 6/8 or 3/4.