# Learn by Doing – Binomial Application

The purpose of this activity is to gain experience making probabilistic decisions using binomial random variables.

## Background

In vitro fertilization is becoming more and more common these days. Suppose each embryo that is implanted has a 20% chance of resulting in a pregnancy that results in delivering a baby. Also, assume that each embryo’s chance of surviving and resulting in a baby is independent of the others.

It is an expensive procedure, so we want to do it only once. We wish to try to find the optimum number of embryos to implant so that the probability of at least 1 child being born is high, but the probability of more than 2 children being born is low. In other words, we want a baby, and we’re willing to have twins, but we don’t want triplets, quadruplets, etc.

Note that unlike the airline flight example, where we needed to control only one probability, in this case there are two probabilities that we wish to control.

The two conditions we’ve outlined mean that we’ll need two probabilities. These are:

- the probability of having at least one child is P(X ≥ 1) which can be found directly or by calculating 1 – P(X = 0)

- the probability of having more than two children is P(X > 2) which can be found directly or by calculating 1 – P(X ≤ 2)

We will let X represent the number of implanted embryos resulting in a baby. It is a binomial random variable with n = number of implanted embryos and p = .20 (the probability that an implanted embryo results in a baby).

It is customary to implant between n = 1 and n = 7 embryos. We have provided a table that contains the two probabilities mentioned in the previous question, for values of n ranging from 1 to 7.

**Note: This table does NOT represent ONE probability distribution**, it is simply a convenient way to summarize the information needed to answer this question.

**We actually needed to use 7 different probability distributions based upon n = the number of embryos.** For each of these distributions, we calculated two probabilities. Check some of these answers for yourself using the online calculator.

# embryos | P(X ≥ 1) | P(X > 2) |
---|---|---|

1 | 0.20 | ≈0 |

2 | 0.36 | ≈0 |

3 | 0.488 | 0.008 |

4 | 0.590 | 0.027 |

5 | 0.672 | 0.058 |

6 | 0.738 | 0.099 |

7 | 0.790 | 0.148 |

http://phhp-faculty-cantrell.sites.medinfo.ufl.edu/files/2013/02/qzLBD_08007.swf

This document is linked from Binomial Random Variables.