Probability problem?

Question

Hi.
Can someone please walk me through the solutions to these 2 problems step by step so I can understand where I went wrong.

1) If a fair coin is tossed 3 times what is the probability of obtaining at most two tails?
2) If a family has three children what is the probability that all three will be female?
Please explain the steps to get a correct solution.
Thank you.

Answer 1

Starting from scratch,
Tossing a fair coin 3 times will have these possible results:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
There are 8, or 2^3 total possibilities, each having a probability of 1/8, or 0.125. You can construct a table as follows:
0H, 3T, 1, 0.125
1H, 2T, 3, 0.375
2H, 1T, 3, 0.375
3H, 0T, 1, 0.125
The probability of at most 2T can be found by adding P(0T), P(1T), and P(2T) or by observing that at most 2T is less than 3T, so that
P(at most 2T) = 1 - P(3T). Either way you arrive at 0.875.
It has also been observed that the numbers and probabilities above can be arrived at by expansion of the binomial (p + q)^3 =
p^3q^0 + (3C1)p^2q^1 + (3C2)p^1q^2 + p^0q^3 wher p is the probability of the single event and q is 1 - p. For larger numbers, this method is easier and quicker than simple counting.

2) For this one, substitute F for H and M for T in the above table. Then P(3F) = 0.125.

Answer 2

1) Not sure on number 1 but here is my attempt:

a coin has two faces: head or tail. It is tossed 3 times, the total possibilities that the coin shows either head or tail is 2^3 = 8

since the statement says at most two tails, we are looking for the probability that the coin shows 1 tails or 2 tails

HHT or TTH

since orders does not matter, for HHT, there are 3 ways you can rearrange the T

for TTH, there are 3! / 2! = 3 ways you can rearrange the T's

so the probability the coin shows one Tail is: 3/8
the probability of the coin shows 2 Tail's is 3/8

so the probability the coin shows at most two tails is:
3/8 + 3/8 = 6/8 = 3/4


same concept.

a child can either be boy or girl. 3 children.
2^3 = 8 (inluding boy and girl)

since there are only 3 childs, there is only one way the three ar girls so the probability is 1/8

Answer 3

1. There is only one configuration (three tails) that goes against the rule of having two tails at most. The probability of having three tails is .5*.5*.5 = .125. The prob. of having at most two tails is 1-.125 = .875

2. Here we have a 50-50 chance in each case (boy or girl).. So using the product rule, .5*.5*.5 = .125

Answer 4

1. 1/2*1/2 =1/4
2.1/2*1/2*1/2=1/8