a student has heard that spinning pennies on a table, rather than flipping them in the air, results in tails…

Explanation:

Step1: Find probability of tails

The probability of a penny landing tails - side up is $p = 0.65$.

Step2: Find probability of all tails

The spins of the 4 pennies are independent events. The probability that all 4 pennies land tails - side up is given by the binomial probability formula $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n = 4$, $k = 4$, and $p = 0.65$. Since $C(4,4)=\frac{4!}{4!(4 - 4)!}=1$, the probability that all 4 pennies land tails - side up is $P(X = 4)=(0.65)^{4}=0.65\times0.65\times0.65\times0.65 = 0.17850625$.

Step3: Find probability of at least one head

The probability of at least one head is the complement of the probability of all tails. Let $A$ be the event of at least one head. Then $P(A)=1 - P(\text{all tails})$. So $P(A)=1-(0.65)^{4}=1 - 0.17850625 = 0.82149375\approx0.8215$.

Answer:

0.8215