To find the relative frequency for the event "heads," we use the formula:
\[ \text{Relative Frequency} = \frac{\text{Number of Heads}}{\text{Total Flips}} \]
For each friend, the total number of flips is 77.
Friend 1:
- Number of Heads = 32
- Total Flips = 77
\[ \text{Relative Frequency for Friend 1} = \frac{32}{77} \approx 0.4156 \]
Rounding to the nearest hundredth:
\[ \text{Relative Frequency for Friend 1} \approx 0.42 \]
Friend 2:
- Number of Heads = 51
- Total Flips = 77
\[ \text{Relative Frequency for Friend 2} = \frac{51}{77} \approx 0.6623 \]
Rounding to the nearest hundredth:
\[ \text{Relative Frequency for Friend 2} \approx 0.66 \]
Friend 3:
- Number of Heads = 55
- Total Flips = 77
\[ \text{Relative Frequency for Friend 3} = \frac{55}{77} \approx 0.7143 \]
Rounding to the nearest hundredth:
\[ \text{Relative Frequency for Friend 3} \approx 0.71 \]
Next, if the friends combine their results where the total number of heads is 138 and the total number of tails is 93, we first need to find the total number of flips:
\[ \text{Total Flips} = \text{Heads} + \text{Tails} = 138 + 93 = 231 \]
Now we can calculate the combined relative frequency for heads:
\[ \text{Relative Frequency for Combined Results} = \frac{138}{231} \approx 0.5974 \]
Rounding to the nearest hundredth:
\[ \text{Relative Frequency for Combined Results} \approx 0.60 \]
Now, if each friend flips a coin 770 times, the expected relative frequency for heads is typically close to the theoretical probability of getting heads with a fair coin, which is:
\[ \text{Theoretical Probability for Heads} = 0.5 \]
So, you would expect the relative frequency for heads to be close to 0.50 when each friend flips the coin enough times.