Simultaneous confidence intervals

How do we show plausible ranges for many estimates at once?

Simultaneous confidence intervals have many applications, including in quality control and opinion polling. The Royal Statistical Society commissioned a survey of MPs, with Savanta ComRes asking:

If you toss a fair coin twice, what is the probability of getting two heads?

As in the previous survey in 2011 with Ipsos MORI, there were seven response options. Parliamentarians could choose: 15%, 25%, 40%, 50%, 75%, ‘Other’…