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To solve this problem, we will use Bayes' Theorem to calculate the probability that the chosen coin is the double-headed one, given that we observed ten consecutive heads. Then, we'll determine the probability that the next flip will also result in a head.
Bayes' Theorem states:
P(Fc∣H=10)=P(H=10)P(H=10∣Fc)⋅P(Fc)
P(H=10)=P(H=10∣F)⋅P(F)+P(H=10∣Fc)⋅P(Fc)
Substitute the values:
P(H=10)=(0.0009765625⋅0.999)+(1⋅0.001)≈0.0019755859375
P(Fc∣H=10)=0.00197558593751⋅0.001≈0.506
Thus, the probability that the chosen coin is the double-headed one given ten consecutive heads is approximately 50.6%.
Using the law of total probability, we calculate:
P(Hnext)=P(Hnext∣Fc)⋅P(Fc∣H=10)+P(Hnext∣F)⋅P(F∣H=10)
Where:
Substitute the values:
P(Hnext)=(1⋅0.506)+(0.5⋅0.494)≈0.753
Therefore, the probability that the next flip will result in a head is approximately 75.3%.