Answer to Question #138721 in Statistics and Probability for nati

Question

IN R studio

# 2. More Coin TossesExperiment: A coin toss has outcomes {H, T}, with P(H) = .6. We do independent tosses of the coin until we get a head.
Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.
Define a random variable "tosses_till_heads" that counts the number of tosses untilwe get a heads. ```{r}```
Use the replicate function, to run 100000 simulations of this random variable. ```{r}```

Use these simulations to estimate the probability of getting a head after 15 tosses. Compare this with the theoretical value computed in the lectures.```{r}```

Compute the probability of getting a head after 50 tosses. What do you notice?```{r}```

Answer 1

Solution

The number of trials before a success follows a geometric distribution. Simulating the number of tosses before the first Head is obtained where the probability of getting a head is 0.6:

```{r}
toss = function() {
  tosses_till_heads = rgeom(1, 0.6)
}
```

In an experiment with 100,000 repetitions, the distribution of the number of tosses until the first head is found as follows:

```{r}
set.seed(5678)


tosses_till_heads = replicate(100000, toss())
mean = mean(tosses_till_heads)
variance = var(tosses_till_heads)
mean
variance
```

The mean and variance of the distribution of trials before getting a head are 0.66203 and 1.0928172 respectively.

```{r}
mean_15 = mean(tosses_till_heads == 15)
mean_15
```

The probability of getting a head after 15 tosses is obtained as 0. The theoretical probability obtained from the pdf

$$p * (1-p)^{n-1} = 0.6 * (1-0.4)^{14} = 0.000001$$

This is consistent with the results from the simulation.

Probability of getting a head after 50 tosses is:

```{r}
mean_50 = mean(tosses_till_heads == 50)
mean_50
```

The probability is 0. Note that the probability of getting a Head after n tosses decreases as n increases. The probability decays to 0 as the number of tosses increases.