Refer to Exercise 1.

(a) Select two wafers, at random and with replacement, from the batch of 10 wafers given in the exercise.

(i) Give the sample space for the experiment that records the doping type of the two wafers and the probability for each outcome.

(ii) Give the sample space of the experiment that records the number of n-type wafers among the two selected and the corresponding probability mass function.

(b) Select four wafers, at random and with replacement, from the batch of 10 wafers given in the exercise.

(i) Give a verbal description of the sample space for the experiment that records the doping type of the four wafers, find the size of the sample space using the R commands G=expand.grid(W1=0:1,W2=0:1,W3=0:1,W4= 0:1); length(G$W1), and give the probability of each outcome.

(ii) Give the sample space of the experiment that records the number of n-type wafers among the four selected and the corresponding PMF using the additional R commands attach(G); table((W1+W2+W3+W4)/4)/length(W1).

(iii) Use the probability mass function to find the probability of at most one n-type wafer among the four selected.

Exercise 1

In electronics, a wafer is a thin slice of semiconductor material used in the fabrication of integrated circuits and other micro-devices. They are formed of highly pure crystalline material, which is doped (i.e., impurity atoms are added) to modulate its electrical properties. The doping is either n-type or p-type. Moreover, the doping is either light or heavy (one dopant atom per 100 million atoms, or per ten thousand atoms, respectively). The following table shows a batch of 10 wafers broken down into the four categories.

One wafer is selected at random. Let E₁ denote the event that the selected wafer is n-type, and E₂ the event that the wafer is heavily doped. Find the probabilities P(E₁), P(E₂), P(E₁ ∩ E₂), P(E₁ ∪ E₂), P(E₁ − E₂), and P((E₁ − E₂) ∪ (E₂ − E₁)).