Q. If a fair coin is tossed 9 times, what is the probability, rounded to the nearest thousandth, of getting at least 7 heads?Answer:
Calculate Probabilities: To solve this problem, we need to calculate the probability of getting exactly 7 heads, exactly 8 heads, and exactly 9 heads, then add these probabilities together. The probability of getting exactly k heads in n tosses of a fair coin is given by the binomial probability formula:P(X=k)=(kn)⋅(pk)⋅((1−p)(n−k))where "n choose k" is the binomial coefficient, p is the probability of getting heads on a single toss (0.5 for a fair coin), and 80 is the random variable representing the number of heads.
Calculate 7 Heads Probability: First, we calculate the probability of getting exactly 7 heads in 9 tosses.Using the binomial coefficient formula, we have:(7)=(7!∗(9−7)!)9!=(7!∗2!)9!=(2∗1)(9∗8)=36(9)Now we calculate the probability:(7)∗(0.57)∗(0.59−7)=36∗(0.57)∗(0.52)P(X=7)=(9)
Calculate 8 Heads Probability: Next, we calculate the probability of getting exactly 8 heads in 9 tosses.Using the binomial coefficient formula, we have:(8)=(8!∗(9−8)!)9!=(8!∗1!)9!=19=9(9)Now we calculate the probability:(8)∗(0.58)∗(0.59−8)=9∗(0.58)∗(0.51)P(X=8)=(9)
Calculate 9 Heads Probability: Finally, we calculate the probability of getting exactly 9 heads in 9 tosses.Using the binomial coefficient formula, we have:(9)=(9!∗(9−9)!)9!=(9!∗0!)9!=1(9)Now we calculate the probability:(9)∗(0.59)∗(0.59−9)=1∗(0.59)∗(0.50)=(0.59)P(X=9)=(9)
Add Probabilities: Now we add the probabilities of getting exactly 7 heads, exactly 8 heads, and exactly 9 heads to find the total probability of getting at least 7 heads.P(X≥7)=P(X=7)+P(X=8)+P(X=9)P(X≥7)=36×(0.57)×(0.52)+9×(0.58)×(0.51)+(0.59)
Perform Calculations: We perform the calculations:P(X≥7)=36×(0.59)+9×(0.59)+(0.59)P(X≥7)=(36+9+1)×(0.59)P(X≥7)=46×(0.59)P(X≥7)=46×(1/512)P(X≥7)=46/512
Simplify and Round: Finally, we simplify the fraction and round to the nearest thousandth: P(X≥7)=51246≈0.0898
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