Question

1. a coin was tossed and a spinner was spun. the spinner has three equal sections numbered 1 through 3. the results are shown in the table below. (table: heads row with 53, 49, 54; tails row with 65, 71, 62) what is the experimental probability that the results will be tails and 3 the next time the coin is tossed and the spinner is spun? a \\(\frac{31}{177}\\) b \\(\frac{9}{59}\\) c \\(\frac{31}{99}\\) d \\(\frac{31}{78}\\)

Explanation:

Step1: Calculate total number of trials

First, we sum up all the frequencies for both heads and tails across all spinner sections.
For heads: \( 53 + 49 + 54 = 156 \)
For tails: \( 65 + 71 + 62 = 198 \)
Total trials: \( 156 + 198 = 354 \)

Step2: Calculate number of favorable trials (tails and spinner 3)

From the table, the number of trials with tails and spinner 3 is 62. Wait, no, wait. Wait, the question is tails and the next time the coin is tossed and spinner is spun, we need the experimental probability of tails and spinner 3? Wait, no, the table: let's re - examine. The spinner has sections 1,2,3. The table rows are Heads, Tails; columns 1,2,3. So for tails: column 1 (spinner 1) is 65, column 2 (spinner 2) is 71, column 3 (spinner 3) is 62. Wait, but the question is "tails and 3 the next time...", so we need the number of times we got tails and spinner 3, which is 62? Wait, no, wait the total number of tails is \( 65 + 71 + 62=198 \), total number of trials is \( (53 + 49 + 54)+(65 + 71 + 62)=156 + 198 = 354 \). Wait, but maybe I misread. Wait, the experimental probability of tails and spinner 3? Wait, no, the question is "the experimental probability that the results will be tails and 3 the next time the coin is tossed and the spinner is spun". So we need the number of times we had tails and spinner 3 divided by total number of trials.

Wait, let's recalculate total trials:
Heads: \( 53+49 + 54=156 \)
Tails: \( 65 + 71+62 = 198 \)
Total trials \( n=156 + 198=354 \)

Number of favorable outcomes (tails and spinner 3): from the table, when spinner is 3 and coin is tails, the frequency is 62? Wait, no, wait the table:

Spinner 1: Heads = 53, Tails = 65

Spinner 2: Heads = 49, Tails = 71

Spinner 3: Heads = 54, Tails = 62

So the number of times we got tails and spinner 3 is 62. Wait, but the answer options: let's check the options. Option A is \( \frac{31}{177} \), which is \( \frac{62}{354}=\frac{31}{177} \) (dividing numerator and denominator by 2). Ah, right! So:

Step1: Find total number of trials

Sum all the values in the table: \( 53 + 49+54 + 65+71 + 62 \)
\(=(53 + 49+54)+(65 + 71+62)=156 + 198 = 354 \)

Step2: Find number of favorable trials (tails and spinner 3)

From the table, the number of trials with tails and spinner 3 is 62.

Step3: Calculate experimental probability

Experimental probability \( P=\frac{\text{Number of favorable trials}}{\text{Total number of trials}}=\frac{62}{354} \)
Simplify the fraction: divide numerator and denominator by 2, we get \( \frac{31}{177} \)

Answer:

A. \( \frac{31}{177} \)