`library(mosaic)`

In Chapter 5, Cartwright refers to Bayes’ Rule. Let’s introduce Bayes’ Rule as a function in R. We need a function of the probability that the outcome is true given the “signal” \(\theta\), or, phrased differently, given some “data” (often called D.) Therefore, we’re looking for something like:

\[Pr(\mbox{Outcome} | \mbox{signal} = \frac{\theta p}{\theta p - (1 - \theta)(1 - p)}\]

```
bayesrule <- function(p, theta) {
(theta * p) / (theta * p + ( 1 - theta)*(1 - p))
}
```

Now that we have a function called `bayesrule()`

, we can specify values for theta and p in order to get `bayesrule()`

to take the values of theta and p as arguments. Let’s use the values Cartwright initially suggests:

```
theta <- 0.9
p <- 0.4
bayesrule(p, theta)
```

`## [1] 0.8571429`

If we wanted to, we could substitute in new values for theta and p, as below:

```
theta <- 0.7
p <- 0.5
bayesrule(p, theta)
```

`## [1] 0.7`

Notice how when p = 0.5, the probability that the outcome is true is simply the accuracy of the signal (here 0.7). So when you “trust” your friend 70% and you are fifty-fifty in your preferences, you’ll be 70% likely to go with what your friend suggests untill you encounter a new signal (or new data).

Let’s put together a

```
ufn1 <- function(w){
(w)^(0.5)
}
wealth <- 0
winnings <- 2
u1 <- ufn1(wealth + winnings)
u2 <- ufn1(wealth)
p <- 0.6
exp.u <- function(p, u1, u2) {
p*u1 + (1 - p)*(u2)
}
exp.u(p, u1, u2)
```

`## [1] 0.8485281`

The gambler’s fallacy: we do not expect many heads (or tails) in a row, even though that could occur totally randomly

```
c1 <- rflip(50)
c1
```

```
##
## Flipping 50 coins [ Prob(Heads) = 0.5 ] ...
##
## T T T T H T H T T T T T H T T H H H H H T T T H T H T H T H T T H
## H H T T T H T H H T H H H T H H T
##
## Number of Heads: 23 [Proportion Heads: 0.46]
```

```
cartwright2 <- data.frame(c("H", "T", "T", "H", "H", "H", "T", "T", "H", "T", "T", "H", "T", "H", "H", "T", "H", "T", "T", "T", "H", "H", "T", "H", "H", "H", "T", "H", "T", "T", "H", "H", "T", "H", "T", "T", "T", "H", "H", "T", "H", "H", "T", "T", "T", "H", "T", "H", "H", "T"))
colnames(cartwright2) <- c("heads")
cartwright2
```

```
## heads
## 1 H
## 2 T
## 3 T
## 4 H
## 5 H
## 6 H
## 7 T
## 8 T
## 9 H
## 10 T
## 11 T
## 12 H
## 13 T
## 14 H
## 15 H
## 16 T
## 17 H
## 18 T
## 19 T
## 20 T
## 21 H
## 22 H
## 23 T
## 24 H
## 25 H
## 26 H
## 27 T
## 28 H
## 29 T
## 30 T
## 31 H
## 32 H
## 33 T
## 34 H
## 35 T
## 36 T
## 37 T
## 38 H
## 39 H
## 40 T
## 41 H
## 42 H
## 43 T
## 44 T
## 45 T
## 46 H
## 47 T
## 48 H
## 49 H
## 50 T
```

`tally(~ heads, data = cartwright2)`

```
##
## H T
## 25 25
```

```
c3 <- rflip(50)
c3
```

```
##
## Flipping 50 coins [ Prob(Heads) = 0.5 ] ...
##
## H T T T H H H H H H T H T T T T H H T H T H H H T T H H T H T H H
## T T H T T T T H T H H H T H H T H
##
## Number of Heads: 27 [Proportion Heads: 0.54]
```

```
GamblersFal <- do(5000) * rflip(10)
histogram( ~ heads, data = GamblersFal, width = 1 )
```

I didn’t end up using this. Ignore it. I’m keeping it here for my notes later.

```
theta.tab <- seq(0.1, 1, by = 0.1)
p.tab <- seq(0.1, 1, by = 0.1)
ptheta <- data.frame(theta.tab, p.tab)
```