Answer:
0.222
Step-by-step explanation:
Given that:
For Type A coins = 3
The probability of head in a type A coin i.e. (p) = 0.4
Then, the probability of getting a tail (q) = 1 - p = 1 - 0.4 = 0.6
For Type B coins = 7
The probability of head in a type B coin i.e. (p) = 0.6
Then, the probability of getting a tail (q) = 1 - p = 1 - 0.6 = 0.4
One person who tosses a coin three times get a probability of obtaining a head twice.
Using, the formula:
=![^nC_r \times p^r \times q^{n-r}](https://tex.z-dn.net/?f=%5EnC_r%20%5Ctimes%20p%5Er%20%5Ctimes%20q%5E%7Bn-r%7D)
For Type A coin;
The probability of getting two heads in three tosses is:
![= ^3C_2 \times 0.4^2 \times 0.6^{1}](https://tex.z-dn.net/?f=%3D%20%5E3C_2%20%5Ctimes%200.4%5E2%20%5Ctimes%200.6%5E%7B1%7D)
![=\dfrac{3!}{2!(3-2)!} \times 0.4^2 \times 0.6^1](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B3%21%7D%7B2%21%283-2%29%21%7D%20%5Ctimes%200.4%5E2%20%5Ctimes%200.6%5E1)
= 0.288
For Type B coin;
The probability of getting two heads in three tosses is:
![= ^3C_2 \times 0.6^2 \times 0.4^{1}](https://tex.z-dn.net/?f=%3D%20%5E3C_2%20%5Ctimes%200.6%5E2%20%5Ctimes%200.4%5E%7B1%7D)
![=\dfrac{3!}{2!(3-2)!} \times 0.6^2 \times 0.4^1](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B3%21%7D%7B2%21%283-2%29%21%7D%20%5Ctimes%200.6%5E2%20%5Ctimes%200.4%5E1)
= 0.432
Since we have two heads out of three tosses, the probability that the coin is type A is = (P) of choosing coin A × (P) of obtaining two heads from three tosses) / total probability of getting two heads from three tosses.
However;
(P) of choosing coin A = 3/10 = 0.3
(P) of choosing coin B = 7/10 = 0.7
∴
Given that, we obtain two head from three tosses, the (P) that the coin type is A is:
![= \dfrac{(0.3 \times 0.288)}{ (0.3 \times 0.288 + 0.7 \times 0.432)}](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B%280.3%20%5Ctimes%200.288%29%7D%7B%20%280.3%20%5Ctimes%200.288%20%2B%200.7%20%5Ctimes%200.432%29%7D)
![= \dfrac{(0.0864)}{ (0.0864 + 0.3024)}](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B%280.0864%29%7D%7B%20%280.0864%20%2B%200.3024%29%7D)
= 0.222