coin a is weighed so that the probability of throwing head is 2/3. coin b is weighed so that the probability of throwing a head is 1/4. coin a is thrown twice and coin b is thrown once.
(I)show that the probability of obtaining exactly 1 head and 2 tails is 13/16
make a list of the possible outcomes of your event
under the headings
a a b
TTH --- (1/3)(1/3)(1/4) = 1/36
THT --- (1/3)(2/3)(3/4) = 6/36
HTT --- (2/3)(1/3)(3/4) = 6/36
total = 1/36 + 6/36 + 6/36) = 13/36
mmmhhh, did I miss something ?
To find the probability of obtaining exactly 1 head and 2 tails when coin A is thrown twice and coin B is thrown once, we can calculate the individual probabilities of obtaining each outcome and then sum them up.
Let's start with coin A:
The probability of throwing head with coin A is 2/3.
The probability of throwing tails with coin A is 1 - (probability of throwing head) = 1 - 2/3 = 1/3.
Now, let's consider the outcomes of throwing coin A twice:
To obtain exactly 1 head and 2 tails, we have 3 possible outcomes:
1. Head, Tail, Tail
2. Tail, Head, Tail
3. Tail, Tail, Head
The probability of each outcome can be calculated by multiplying the probabilities of getting the specific result for each coin toss. Since each toss is independent, we can multiply the individual probabilities.
For Outcome 1:
Probability = (Probability of Head) * (Probability of Tail) * (Probability of Tail)
= (2/3) * (1/3) * (1/3)
= 2/27
For Outcome 2:
Probability = (Probability of Tail) * (Probability of Head) * (Probability of Tail)
= (1/3) * (2/3) * (1/3)
= 2/27
For Outcome 3:
Probability = (Probability of Tail) * (Probability of Tail) * (Probability of Head)
= (1/3) * (1/3) * (2/3)
= 2/27
Now, we sum up the probabilities of these three outcomes to get the probability of obtaining exactly 1 head and 2 tails with coin A:
Probability = 2/27 + 2/27 + 2/27
= 6/27
= 2/9
So, the probability of obtaining exactly 1 head and 2 tails with coin A is 2/9.
Next, let's consider coin B:
The probability of throwing head with coin B is 1/4.
The probability of throwing tails with coin B is 1 - (probability of throwing head) = 1 - 1/4 = 3/4.
Now, we have only one outcome for throwing coin B as it is thrown only once:
Head, or Tail (as there is no condition for the second toss)
The probability of this outcome can be calculated by multiplying the probabilities of getting the specific result for coin B:
Probability = (Probability of Head) * (Probability of Tail)
= (1/4) * (3/4)
= 3/16
So, the probability of obtaining exactly 1 head and 2 tails with coin B is 3/16.
Finally, to find the overall probability of obtaining exactly 1 head and 2 tails, we need to consider the probabilities of these outcomes together.
Probability = Probability of obtaining exactly 1 head and 2 tails with coin A * Probability of obtaining exactly 1 head and 2 tails with coin B
= (2/9) * (3/16)
= 6/144
= 1/24
= 4/96
To simplify, we can multiply the numerator and denominator of 4/96 by 4:
Probability = (4/4) * (1/24)
= 1/6 *1/24
= 1/144
Hence, the probability of obtaining exactly 1 head and 2 tails is 1/144.
To calculate the probability of obtaining exactly 1 head and 2 tails, we need to consider the possible outcomes for each coin toss and then calculate the probability of each outcome occurring.
Coin A:
Since the probability of throwing a head with Coin A is 2/3, the probability of throwing a tail is 1 - 2/3 = 1/3.
Coin B:
Similarly, the probability of throwing a head with Coin B is 1/4, so the probability of throwing a tail is 1 - 1/4 = 3/4.
Now, let's consider the possible outcomes for the three coin tosses:
1. Head-Head-Tail (HHT)
The probability of this outcome is (2/3) * (2/3) * (3/4) = 4/9 * 3/4 = 12/36 = 1/3.
2. Head-Tail-Head (HTH)
The probability of this outcome is (2/3) * (1/3) * (3/4) = 2/9 * 3/4 = 6/36 = 1/6.
3. Tail-Head-Head (THH)
The probability of this outcome is (1/3) * (2/3) * (3/4) = 2/9 * 3/4 = 6/36 = 1/6.
Adding up the probabilities of these three outcomes, we get (1/3) + (1/6) + (1/6) = 2/3.
Thus, the probability of obtaining exactly 1 head and 2 tails is 2/3.
Therefore, the statement that the probability of obtaining exactly 1 head and 2 tails is 13/16 is incorrect.
Please let me know if I can help you with anything else.