What is the probability of obtaining exactly three heads in four flips of a coin, given that at least two are heads? (Enter the probability as a fraction.)
I keep getting 2/3 but its wrong. Help!!
two are already decide. You have two more flips. What is the chance of getting one heads and one tails out of those two flips ?
You have four possibilities
HH
HT
TH
TT
two of those four work, TH and HT
so I get 1/2
To find the probability of obtaining exactly three heads in four flips of a coin, given that at least two are heads, we need to consider the different possible scenarios.
First, let's look at the possible ways of getting at least two heads in four flips:
- HHHT
- HHTH
- HTHH
- THHH
- HHHH
Out of these, only three scenarios have exactly three heads (HHHT, HHTH, HTHH). Therefore, the probability of getting exactly three heads in four flips, given that at least two are heads, is 3 out of 5.
So, the correct probability is 3/5.
To solve this problem, we need to determine the probability of obtaining exactly three heads in four flips of a coin, given that at least two are heads.
First, let's consider the probability of getting two heads. This can occur in three different ways: HHT, HTH, or THH, where H represents a head and T represents a tail. Each of these scenarios has a probability of (1/2)^2 * (1/2)^1 = 1/8.
Now, let's consider the probability of obtaining exactly three heads. This can occur in four different ways: HHHK, HHKH, HKHH, or KHHH, where K represents a tail. Each of these scenarios has a probability of (1/2)^3 * (1/2)^1 = 1/16.
Next, let's find the probability of at least two heads occurring in four flips. This includes the probability of obtaining two heads (1/8) and the probability of obtaining three heads (1/16). Therefore, the total probability of at least two heads is 1/8 + 1/16 = 3/16.
Finally, we need to find the conditional probability of obtaining exactly three heads, given that at least two are heads. We can use the formula for conditional probability:
P(A | B) = P(A and B) / P(B)
In this case, A represents obtaining exactly three heads and B represents at least two heads. Therefore, we have:
P(exactly three heads | at least two heads) = P(exactly three heads and at least two heads) / P(at least two heads)
P(exactly three heads | at least two heads) = (1/16) / (3/16)
P(exactly three heads | at least two heads) = 1/3
Therefore, the probability of obtaining exactly three heads in four flips of a coin, given that at least two are heads, is 1/3.